Saratov 2010
Compiled by: R.P. Volkov, lecturer at FGOU SPO "SGPPC named after Yu. A. Gagarin"
Internal reviewer: O.G. Stegalkina - lecturer
FGOU SPO "SPPK named after Yu.A. Gagarin"
Guidelines for solving problems and performing independent work on the course "Physical and chemical foundations for the development and cessation of combustion in a fire" for students of all forms of education in the specialty "Fire Safety".
The guidelines consider examples of solving typical problems in the section “Fundamentals of combustion processes. Material and heat balance of combustion processes" discipline "Physical and chemical bases of development and cessation of combustion in a fire"; given options for tasks for independent problem solving.
Printed in the printing house of the FGOU SPO “SGPPC named after Yu.A. Gagarin"
INTRODUCTION
Guidelines for solving problems and performing independent work on the section “Fundamentals of combustion processes. Material and heat balance of combustion processes" of the discipline "Physical and chemical bases of development and cessation of combustion in a fire" are designed to train fire safety engineers within the framework of the work program of the discipline "Physical and chemical bases of development and cessation of combustion in a fire" in the specialty 280104.
Methodological instructions for solving problems are drawn up in full accordance with the State Educational Standard for Higher Professional Education, taking into account the specifics of the professional activities of the State Fire Service employees. Tasks are designed to consolidate the theoretical course and methods of practical calculations in this section of the discipline. Guidelines will help students master the material of the discipline being studied, which is necessary for the successful work of a fire safety engineer in any field of his activity.
The guidelines include: brief theoretical provisions, general provisions for calculating the material and heat balance of combustion processes of gaseous and condensed substances, the nature of the flame glow, combustion temperature, as well as a large number of examples of solving typical problems and reference information necessary for solving problems.
The structure and content of guidelines for solving problems provide for the possibility of self-working out by trainees of the material for each section of the discipline.
Starting to study the course, it is necessary to imagine that the basis of all phenomena occurring in a fire is the combustion process. Knowledge of the essence of this phenomenon, the laws of combustion, the mechanisms and methods for its termination are necessary for the successful work of a fire safety engineer in any field of his activity.
1. Write structural formulas, draw up equations for the reactions of combustion of combustible substances in air and calculate stoichiometric coefficients.
1.1. amylbenzene, abietic acid, allylamine;
1.2. amyl diphenyl, adipic acid, allyl isothiocyanate;
1.3. amylene, acrylic acid, alnapht;
1.4. amylnaphthalene, allyl acetate, altax;
1.5. amyltoluene, allylidene diacetate, amylamine;
1.6. anthracene, allyl caproate, amyl nitrate;
1.7. acenaphthene, allyl alcohol, amyl nitrite;
1.8. acetylene, amyl acetate, amyl sulfide;
1.9. benzene, amyl butyrate, amyl trichlorosilane;
1.10. butylbenzene, amylxylyl ether, amylchloronaphthalene;
1.11. butylcyclohexane, amillaurate, aminalon;
1.12. butylcyclopetane, amyl methyl ketone, amino azo dye;
1.13. hexadecane, amyloleate, aminocaproic acid;
1.14. hexane, amyl salicylate, aminopelargonic acid;
1.15. hexylcyclopentane, amyl stearate, aminocyclohexane;
1.16. heptadecane, amylphenyl methyl ether, ampicillin;
1.17. heptane, amnlphenyl ether, anginine;
1.18. decane, amyl formate, aniline;
1.19. diamylbenzene, anisole, antrimid;
1.20. diamylnaphthalene, acetal, atophane;
1.21. divinylacetylene, acetaldehyde, aceclidine;
1.22. dihydrocyclopentadiene, acetylacetone, acetanilide;
1.23. diisobutylene, acetysalicylic acid, acetyl chloride;
1.24. diisopropylbenzene, acetyltributylcitrate, acetoacetanilide;
1.25. dimethylenecyclobutane, acetomethoxane, acetonitrile;
1.26. ditolylmethane, acetone, acetoxime;
1.27. diphenyl, acetonylacetone, acetoethylamide;
1.28. diphenylmethane, acetopropyl alcohol, benzamide;
1.29. diethylcyclohexane, acetoacetic ether, benzyldiethylamine;
1.30. dodecane, acetophenone, benzylthiol;
1.31. isobutylbenzene, benzaldehyde, benzyl chloride;
1.32. isobutylcyclohexane, benzantrone, benzyl cyanide;
1.33. isooctane, benzhydrol, benzimidazole;
1.34. isopentane, benzyl acetate, sodium benzoate;
1.35. isoprene, benzyl benzoate, benzoyl chloride;
1.36. isopropenylbenzene, benzyl salicylate, benzoxazolone;
1.37. isopropylacetylene, benzyl cellosolve, benzene sulfazide;
1.38. methylcyclohexane, benzyl ethyl ether, benzenesulfamide;
1.39. methylcyclopentane, benzylsuccinic acid, benzenesulfonic acid;
1.40. octyltoluene, methoxybutyl acetate, benzonitrile.
2. Write structural formulas and determine during the combustion of which combustible substance the greater number of moles of combustion products will be released?
2.1. benzophenone and benzophenone tetracarboxylic acid;
2.2. borneol and butanal;
2.3. butanoic acid and butyl acetate;
2.4. butylacetylricinoleate and butylacetoacetate
2.5. butylbenzyl sebacate and butyl benzoate;
2.6. butyl butyrate and butyl vinyl ether;
2.7. butyl glycol and butyl glycol acetate;
2.8. butylglycide ether and butyldiethyl adipate;
2.9. butylisovalerate and butylcapronate;
2.10. butyl carbitol and butyl lactate;
2.11. butyl laurate and butyl methacrylate;
2.12. butyl methyl ketone and butyl oleate;
2.13. butylpropionate and butylricinooleate;
2.14. butyl stearate and butylphenyl ether;
2.15. butyl formate and butylethyl acetaldehyde;
2.16. butyl ethyl ketone and butyl ethyl ether;
2.17. valeric acid and valeric aldehyde;
2.18. vanillin and vetiveryl acetate;
2.19. vetiver alcohol and vetinyl acetate;
2.20. vetinone and vinylallyl ether;
2.21. vinyl acetate and vinyl butyrate;
2.22. vinyl isobutyl ether and vinyl isooctyl ether;
2.23. vinyl isopropyl ether and vinyl crotonate;
2.24. vinyl methyl ketone and vinyl oxyethyl methacrylate;
2.25. vinyl octadecyl ether and vinyl propionate;
2.26. vinyl trimethyl nonyl ether and vinyl ethyl ether;
2.27. vinyl ethyl ether and tartaric acid;
2.28. vitamin A (acetate) and vitamin C;
2.29. gallic acid and hexanal;
2.30. hexanoic acid and hexyl acetate;
2.31. hexyl butyrate and hexyl diethyl hexahydrophthalate;
2.32. hexyl methacrylate and hexyl methyl ketone;
2.33. hexyl alcohol and hexyl propionate;
2.34. hexyl formate and hexyl cellosolve;
2.35. heliotropin and heptadecyl alcohol;
2.36. heptanal and heptylacetate;
2.37. heptyl butyrate and heptyl diphenyl ketone;
2.38. heptyl isobutyl ketone and heptyl methyl ketone;
2.39. heptyl alcohol and heptyl propionate;
2.40. heptyl formate and hydroquinone.
DECISION.
1. We compose the equations for the reactions of combustion of combustible gases of the mixture in air:
C 2 H 2 + 2.5 (O 2 + 3.76 N 2) \u003d 2 CO 2 + H 2 O + 2.5*3.76N2 ,
C 3 H 8 + 5 (O 2 + 3.76 N 2) \u003d 3 CO 2 + 4 H 2 O + 5*3.76N2.
2. Calculate the theoretical volumes of air and combustion products during complete combustion of 1 m 3 of the gas mixture (formulas 8 and 15):
3. Calculate the actual volumes of air and combustion products, taking into account a 40% excess of air (α = 1.4).
4. Since the volume of the combustible mixture was 10 m 3, the actual volumes of air and combustion products will be 176.7 and 192.9 m 3, respectively.
ANSWER: The combustion of 10 m 3 of a complex gas mixture requires 176.7 m 3 of air, while 192.9 m 3 of combustion products are formed.
EXAMPLE:Determine the volumes of air and combustion products during the combustion of 2 kg of a combustible substance having an elemental composition: C = 50%; H = 10%; N = 10%; ash = 12%; moisture = 18%. Assume that the air and combustion products are under normal conditions.
DECISION:
1. To solve the problem, we use formulas (9) and (16).
When burning 2 kg of combustible substance, 14.34 and 16.14 m 3 of air and combustion products are formed, respectively.
ANSWER: When burning 2 kg of combustible substance, 14.34 m 3 of air is consumed and 16.14 m 3 of combustion products are formed.
TASKS FOR INDEPENDENT SOLUTION
1. Determine the volume of air required for the combustion of 50 m 3 of acetylene at α=1, 7.
2. Determine the volumes of air, combustion products and the percentage of combustion products in 2 m 3 of ethane. Take the temperature of the combustion products to be 1200 K, pressure 101.3 kPa, excess air α=1.2.
3. Determine the volume of air required for the combustion of 15 m 3 of butane at a temperature of 10С and a pressure of 750 mm Hg. Art., if combustion proceeds with an excess air coefficient equal to 1.4 (α=1.4).
6. Calculate the amount of amylbenzene that can burn in a closed room with a volume of 200 m 3 if combustion stops at a residual oxygen content of 12%. The initial temperature in the room is 24 ° C, the pressure is 98 kPa.
7. Determine how much butyl acetate can burn in a room with a volume of 200 m 3 if its combustion stops at an oxygen content of the air equal to 13.8% (normal conditions).
8. Determine the volumes of combustion products and air during the combustion of 7 kg of hexane. The combustion process proceeded at a temperature of 33°C and a pressure of 730 mm. rt. Art. The temperature of the combustion products is assumed to be 1300 K.
9. Determine the volumes of combustion products and air during the combustion of 11 kg of acetone. The combustion process proceeded at a temperature of 30 about C and a pressure of 720 mm Hg. Art. The temperature of the combustion products is assumed to be 1300 K.
10. Determine the volume of combustion products and air during the combustion of 17 kg of toluene. The combustion process proceeded at a temperature of 30 about C and a pressure of 745 mm Hg. Art. The temperature of the combustion products is assumed to be 1100 K.
11. Calculate the volume of air and the volume of combustion products during complete combustion of 6 kg of cellulose, consisting of 80% carbon, 13% hydrogen and 7% oxygen, if combustion occurs at a temperature of 25 ° C and a pressure of 95 kPa. The excess air coefficient is 1.4.
12. Determine the volume of air required for the combustion of 6 kg of diethyl ether at a temperature of 15 ° C and a pressure of 750 mm Hg. Art. The excess air coefficient was 1.3.
13. Determine how much benzene burned in a closed room with a volume of 180 m 3, if it is known that its combustion stopped when the oxygen content in the air was 14.6%. The temperature before the fire was 19 o C and pressure 100 kPa.
15. Determine the coefficient of excess air if 212 m 3 of air is consumed for the combustion of 8 kg of ethyl acetate at a temperature of 25 ° C and a pressure of 760 mm Hg. Art.
16. Calculate the excess air coefficient and the percentage of carbon dioxide in the combustion products if 70 m 3 of air is consumed for the complete combustion of 4 kg of ethyl propyl ether (C 5 H 12 O) at a temperature of 22 ° C and a pressure of 92 kPa.
17. 3 kg of acrolein burns at a temperature of 21 ° C and a pressure of 98 kPa. Calculate the volume of air that has passed into the combustion products and the percentage of water in them if combustion proceeds with an excess of air (the excess air coefficient is 1, 2).
20. Calculate the volume of a gas mixture consisting of 45% butane, 30% methane, 20% acetylene and 5% oxygen, if 80 m 3 of air is consumed for its combustion under normal conditions. The excess air coefficient is 1.6.
22. Calculate the volume of air and the volume of combustion products during complete combustion of 7 m 3 of a gas mixture consisting of 57% hydrogen, 18% carbon monoxide and 25% methane, if combustion occurs with excess air (the excess air coefficient is 1.3).
23. Calculate the volume of air and the volume of combustion products during the complete combustion of 6 kg of ceresin, consisting of 80% carbon, 15% hydrogen and 5% oxygen, if combustion occurs at a temperature of 25 ° C and a pressure of 95 kPa. The excess air coefficient is 1.5.
25. Determine the volume and composition of combustion products (in vol.%) of the gas mixture (Table 4), if combustion occurs at an excess air coefficient α (see Table 4).
Table 4
| Composition of the mixture, % | Job number | |||||||||
| 25.1 | 25.2 | 25.3 | 25.4 | 25.5 | 25.6 | 25.7 | 25.8 | 25.9 | 25.10 | |
| carbon monoxide | - | - | - | - | - | - | ||||
| Hydrogen | - | - | - | - | - | - | - | |||
| Methane | - | - | - | - | - | - | ||||
| Ethane | - | - | - | - | - | -- | ||||
| Propane | - | - | - | - | - | - | - | - | - | |
| Butane | - | - | - | - | - | - | - | - | ||
| Ethylene | - | - | - | - | - | |||||
| propene | - | - | - | - | - | - | - | - | ||
| Acetylene | - | - | - | - | - | - | - | |||
| Carbon dioxide | - | - | ||||||||
| Nitrogen | - | - | - | - | ||||||
| Oxygen | - | - | - | |||||||
| α | 1,2 | 1,3 | 1,1 | 1,2 | 1.2 | 1,2 | 1,4 | 1,1 | 1,3 |
Table 5
| job number | Substance | Elemental composition of matter | t o C | Mass of substance, kg | |||||
| C | H | O | S | W | ash | ||||
| 26.1 | Ceresin | ||||||||
| 26.2 | Coal | ||||||||
| 26.3 | Wood | ||||||||
| 26.4 | Petrol | ||||||||
| 26.5 | Oil | ||||||||
| 26.6 | fuel oil | ||||||||
| 26.7 | Kerosene | ||||||||
| 26. 8 | oil shale | ||||||||
| 26.9 | Coal | ||||||||
| 26.10 | Anthracite | 0,2 | 5,8 |
27. Determine the nature of the glow of the flame of ethylbenzene.
28. Determine the nature of the glow of the flame of acetic acid.
29. Determine the nature of the hexane flame glow.
30. Determine the nature of the glow of the flame of amyl alcohol.
31. Determine the nature of the glow of the butane flame.
32 . Determine the nature of the glow of the benzene flame.
CALCULATION OF COMBUSTION TEMPERATURE
The combustion temperature is understood as the maximum temperature to which the combustion products are heated. In engineering and fire fighting, a distinction is made between theoretical, calorimetric, adiabatic, and actual combustion temperatures.
Theoretical combustion temperature is the temperature at which the released heat of combustion of a mixture of stoichiometric composition is spent on heating and dissociation of combustion products. In practice, the dissociation of combustion products begins at temperatures above 2,000 K.
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Calorimetric combustion temperature is the temperature that is reached during the combustion of a stoichiometric combustible mixture with an initial temperature of 273 K and in the absence of losses to the environment.
Adiabatic combustion temperature is the temperature of complete combustion of mixtures of any composition in the absence of heat losses to the environment.
Actual combustion temperature is the combustion temperature achieved in a real fire. It is much lower than the theoretical, calorimetric and adiabatic, because in real conditions, up to 40% of the heat of combustion is usually lost to radiation, underburning, heating of excess air, etc.
Experimental determination of the combustion temperature for most combustible substances presents significant difficulties, especially for liquids and solid materials. However, in a number of cases, the theory makes it possible to calculate the combustion temperature of substances with sufficient accuracy for practice, based only on knowledge of their chemical formula, the composition of the initial combustible mixture and combustion products.
In the general case, the following dependence is used for calculations (approximate, since C p \u003d f (T)):
Q pg \u003d V pg * C p * T g,
where Q pg - enthalpy of combustion products;
V pg - the amount of combustion products, m 3 /kg;
C p - the average volumetric heat capacity of the mixture of combustion products in the temperature range from T 0 to T g, kJ / (m 3 * K);
T g - combustion temperature, K.
The enthalpy of combustion products is determined from heat balance equations:
Q pg \u003d Q H + Q ref - Q sweat, (24)
Q sweat = Q u + Q nedo + Q dis with , (25)
where Q use is the heat of evaporation;
Q sweat- heat loss due to radiation, underburning and dissociation combustion products.
Depending on the type of heat losses taken into account in the combustion zone (for radiation, underburning, dissociation of combustion products), one or another temperature is calculated.
During kinetic combustion of gas-vapor-air mixtures, heat losses from the combustion zone are negligibly small, therefore, for these mixtures, the actual combustion temperature is close to adiabatic, which is used in fire engineering calculations.
It is very difficult to determine the average heat capacity of a mixture of combustion products. Roughly, the enthalpy of a mixture of combustion products can be expressed as the sum of the enthalpies of its components:
Qpg =Σ (V pg) i (С р) i*T g, (26)
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where (V pg) i is the amount of the i-th component of combustion products;
C p is the average volumetric heat capacity of the ith component at T g and constant pressure
T g- combustion temperature.
When calculating the combustion temperature, the value is used Q n(lower calorific value), since at the combustion temperature water is in a gaseous state.
The values of the net calorific value of a substance (heat effect of a chemical reaction) are given in the reference literature, and can also be calculated from the corollary of Hess's law:
Q n \u003d (Σ ΔН i *n i) prod - (ΣΔН i * n i) ref, where (27)
ΔН i is the heat of formation of the i-th substance,
n i is the number of moles of the i-th substance.
According to corollary of Hess's law the thermal effect of a chemical reaction is equal to the difference between the sums of the heats of formation of the reaction products and the heats of formation of the starting materials. Recall from the course of chemistry that the heat of formation of simple substances (oxygen, nitrogen, etc.) is zero.
For example, let's calculate the heat of combustion (thermal effect) of ethane:
C 2 H 6 + 3.5*(O 2 + 3.76N 2) \u003d 2 CO 2 + 3 H 2 O + 3.76 *3.5N2.
The lower calorific value, according to the Hess consequence, is equal to:
Q n \u003d ΔH CO 2 * n CO 2 + Δ H H 2 O * n H 2 O - ΔH C 2 H 6 * n C 2 H 6 (28)
Substituting the values of the heat of formation of CO 2 , H 2 O, C 2 H 6 from reference data, determine the lower calorific value of ethane.
When a mixture of individual substances is burned, the heat of combustion of each component is first determined, and then they are summed up, taking into account the percentage of each combustible substance in the mixture:
If the fuel is a complex substance and its elemental composition is given in mass percent, then the Mendeleev formula is used to calculate the heat of combustion:
Q n c m \u003d 339.4 * C + 1257 * H-108.9 (O-N-S) -25 (9 * H + W), kJ / kg (30)
where C, H, O, N, S is the percentage of this element in the combustible substance;
W– moisture content in mass. %.
To calculate the combustion temperature, we compose the heat balance equation, assuming that the heat released as a result of combustion heats the combustion products from the initial temperature T 0 up to temperature T Mr.:
Q n (1-η) \u003d Σs rpg i * V pg i (T g -T 0)
Where η – heat loss coefficient (share of heat loss due to radiation, as well as as a result of incomplete combustion);
with RPG i – heat capacity of the i-th combustion product at constant pressure, kJ/molK;
V pg i - the volume of the i-th product of combustion, m 3.
Calculation of the volume of combustion products ( CO 2, H 2 O, SO 2, N 2) is carried out according to the following formulas:
From the heat balance equation:
The difficulty in determining the combustion temperature using this formula is that the heat capacity of a gas depends on temperature. Since gases are heated by temperature T 0 up to temperature T g, then in formula (36) it is necessary to substitute the average value of the heat capacity in this temperature range. But the combustion temperature is unknown to us and we want to find it. In this case, you can do the following. The average value of the combustion temperature of most substances in the air is approximately 1500 K. Therefore, with a small error in determining T g for calculations, we can take the average value of the heat capacity in the temperature range 273–1500 K. These values for the main combustion products are given in Table. 6.
Table 6
Average heat capacities of the main combustion products in the temperature range 273–1500 o C
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The average value of the heat capacity of some gaseous substances in various temperature ranges is also given in Table. III applications.
Consider examples of solving problems for calculating the combustion temperature.
TASKS FOR INDEPENDENT SOLUTION
1. In which case, under fire conditions, during the combustion of butane, more heat will be released: with complete combustion or incomplete, proceeding by the reaction C 4 H 10 + 4.5O 2 →4CO + 5H 2 O. The answer must be confirmed by calculation using Hess's law.
2. Calculate the heat of formation of acetylene from the elements if its heat of combustion is 1411.2 kJ/mol.
3. Determine the heat of combustion of 12 kg of benzene, if the heat of its formation is 82.9 kJ/mol, the heat of formation of carbon dioxide is 396.9 kJ/mol, the heat of formation of water vapor is 242.2 kJ/mol.
4. Determine the heat of formation of pimelic acid (C 7 H 12 O 4), if the heat of combustion is 3453.5 kJ / mol.
5. Determine the heat of combustion of salicylic acid if the heat of its formation is 589.5 kJ/mol.
6. Calculate the heat of formation of methane if the combustion of 10 g of it under standard conditions releases 556.462 kJ of heat.
7. Determine the heat of combustion of benzyl alcohol (C 7 H 8 O), if the heat of its formation is 875.4 kJ / mol.
8. During the formation of octane, 208.45 kJ / mol of heat is released from the elements. Calculate its heat of combustion.
9. The heat of formation of acetone is -248.28 kJ/mol. Determine its heat of combustion and the amount of heat that is released during the combustion of 30 g of the substance.
11. Determine the calorific value of sulfophenylhydrazine (C 6 H 8 O 3 N 2 S) taking into account losses due to water evaporation. The moisture content of the substance is 20%.
12. Determine the calorific value of 4, 4 / -diaminodiphenylsulfone (C 12 H 12 O 2 N 2 S) without taking into account moisture evaporation losses.
13. Determine the heat of combustion of 4, 6-dimethylhexahydro-1, 3, 5-triazinethion-2 (C 5 H 9 N 3 S) according to the formulas of D. I. Mendeleev.
14. Determine the heat of combustion of diaminomesitylene-6-sulfonic acid (C 9 H 14 O 3 N 2 S), if the moisture content in the substance is 35%.
15. Determine the lowest calorific value of wood composition: C - 41.5%; H - 6%; O - 43%; N, 2%; W– 7.5%.
16. Determine the theoretical combustion temperature of acetone using the average heat capacities.
17. Determine the theoretical combustion temperature of pentane using the average heat capacities.
18. Determine the theoretical combustion temperature of octane using average heat capacities.
19. Determine the theoretical combustion temperature of benzene using the average heat capacities.
20. Using the method of successive approximations, calculate the adiabatic temperature of the combustion of propanol.
21. Calculate the combustion temperature for a stoichiometric mixture of combustible substance with air (Table 7).
Table 7
22. Using the method of successive approximations, calculate the actual combustion temperature of a combustible substance (Table 8), if combustion occurs at an excess air coefficient α, and the proportion of heat loss by radiation is η.
Table 8
| job number | Substance name | Elemental composition of matter, wt. % | α | η | ||||||
| C | H | O | S | N | W | ash | ||||
| 22.1 | Anthracite | 0,5 | 1,0 | 21,5 | 1,1 | 0,2 | ||||
| 22.2 | Oil shale | 24,2 | 1,8 | 4,5 | 3,0 | 2,0 | 39,5 | 1,2 | 0,3 | |
| 22.3 | Kerosene | 13,7 | 0,3 | - | - | - | 1,3 | 0,4 | ||
| 22.4 | Petrol | 8,0 | 5,0 | - | 2,0 | 1,4 | 0,3 | |||
| 22.5 | Salt oil | 86,0 | 12,0 | 1,2 | 0,8 | - | - | - | 1,5 | 0,2 |
| 22.6 | fuel oil | - | - | 1,6 | 0,3 | |||||
| 22.7 | Wood | - | - | 1,7 | 0,4 | |||||
| 22.8 | Coal | - | 1,8 | 0,3 | ||||||
| 22.9 | Ceresin | - | - | - | - | 1,7 | 0,2 | |||
| 22.10 | Oil shale | 1,6 | 0,3 |
23. Determine the theoretical combustion temperature of the rubber compound: With = 80 %, H= 15%, S = 2%, O = 1%, N = 2 %.
24 . Determine the actual combustion temperature of the composition paper: C \u003d 55%, H \u003d 25%, N \u003d 3%, O \u003d 15%, H 2 O \u003d 2%, if the heat loss due to underburning was η X=0.15, due to radiation η izl=0,20.
25. Determine the actual combustion temperature of the composition plastic: C = 70%, H = 20%, N = 5%, O = 2%, non-combustible components (fillers) amounted to 3% /, if heat loss due to underburning amounted to η X=0.20, due to radiation η izl=0.25. Excess air coefficient α = 1, 4.
APPENDIX
List of accepted designations
n is the number of moles of the substance;
β is the stoichiometric coefficient;
V to theor- theoretically necessary for combustion, m 3;
V in d- the actual (practical) volume of air that went to combustion, m 3;
V pg t- theoretical volume of combustion products, m 3;
R– gas pressure, Pa;
P 0– initial (atmospheric) pressure, Pa;
T is the temperature of the substance, K;
Q is the amount of heat, J;
Vi- the volume of the i-th gaseous substance, m 3, kmol;
α - coefficient of excess air;
m is the mass of the substance, kg;
M is the mass of one kmole of a substance, kg/kmol;
Q n– lower calorific value of substance, kJ/mol, kJ/kg;
H i- enthalpy of the i-th substance, kJ / mol, kJ / m 3;
T g– combustion temperature, K;
cpi is the heat capacity of the i-th gas at constant pressure, kJ/mol*K; kJ/m3;
η is the heat loss coefficient.
Table I
Basic physical constants of some gases
A traveler from Moscow wants to visit four cities of the Golden Ring of Russia: Vladimir, Yaroslavl, Suzdal and Rostov the Great. The travel agency offers routes with visits to some cities of the Golden Ring. Information about the cost of tickets and routes are presented in the table.
What routes should a traveler choose to visit all four cities and spend less than 5,500 rubles? Specify any one set of route numbers in your answer
without spaces, commas and other additional characters.
To transport 42 tons of cargo over 1200 km, you can use the services of one of three carrier companies. The cost of transportation and the carrying capacity of vehicles of each carrier are indicated in the table.
How many rubles will you have to pay for the cheapest transportation?
Using the table, gather at least one group in which translators together speak all four languages: English, German, Spanish and French, and the total cost of their services does not exceed 12,000 rubles per day. In your answer, indicate any one set of translator numbers without spaces, commas, or other additional characters.
20. Mikhail decided to visit the Amusement Park. Ticket details
rides are presented in the table. Some tickets allow you to visit two attractions at once.
Using the table, select tickets so that Mikhail visits all four attractions: the Ferris wheel, the fear room, the laughter room, the circuit, and the total cost of tickets does not exceed 750 rubles. In your answer, indicate any one set of ticket numbers without spaces, commas, and other additional characters.
21. The telephone company provides three service plans to choose from.
The subscriber assumes that the total duration of calls will be 700 minutes per month, and based on this, he chooses the cheapest tariff plan. How many rubles will the subscriber have to pay per month if the total duration of calls is indeed 700 minutes?
22. When building a house, the company uses one of the types of foundation: stone or concrete. For a stone foundation, 9 tons of natural stone and 8 bags of cement are needed. For a concrete foundation, 6 tons of crushed stone and 60 bags of cement are needed. A ton of stone costs 1,700 rubles, crushed stone costs 770 rubles per ton, and a bag of cement costs 240 rubles. How many rubles will the material for the foundation cost if you choose the cheapest option?
The city park has five attractions: a carousel, a Ferris wheel, an autodrome, "Chamomile" and "Merry Shooting Gallery". The box office sells six types of tickets, each of which allows you to visit one or two attractions. Information about the cost of tickets is presented in the table.
What tickets should Andrey buy to visit all five attractions and spend no more than 900 rubles? In your answer, indicate any one set of ticket numbers without spaces, commas, and other additional characters.
The tourist selects excursions. Information about excursions is presented in the table.
| Excursion number | Visited objects | Cost, rub.) |
| The park | ||
| Fortress, park | ||
| Fortress | ||
| Fortress, country palace | ||
| Museum of Painting | ||
| Country Palace, Museum of Painting |
Using the table, select a set of excursions so that the tourist visits four objects: a fortress, a country palace, a park and a museum of art, and the total cost of excursions does not exceed 650 rubles. In your answer, indicate any one set of tour numbers without spaces, commas, and other additional characters.
25. The Internet provider offers three tariff plans.
The user assumes that his traffic will be 650 MB per month,
and based on this chooses the cheapest tariff plan. How many rubles will a user have to pay per month if his traffic is really 650 MB?
To service an international seminar, it is necessary to assemble a group of translators. Information about the candidates is presented in the table.
Using the table, gather at least one group in which translators together speak all four languages: English, German, Spanish and French, and the total cost of their services does not exceed 12,000 rubles per day. In your answer, indicate any one set of translator numbers without spaces, commas, or other additional characters.
27. A rating agency ranks electric hair dryers
based on the average price P (in rubles), as well as indicators of functionality F, quality Q and design D. Rating R is calculated by the formula R=3(F+Q)+D−0.01P. The table shows the prices and performance of four models of hair dryers.
The client wants to rent a car for a day for a trip of 600 km. The table shows the characteristics of three cars and the cost of their rental.
In addition to the rental, the client is obliged to pay for fuel for the car for the entire trip. The price of diesel fuel is 25 rubles per liter, gasoline - 35 rubles per liter, gas - 20 rubles per liter. How many rubles will the client pay for rent and fuel if he chooses the cheapest option?
29. An independent expert laboratory determines the rating of meat grinders
based on a value coefficient equal to 0.01 of the average price P (in rubles), functionality F, quality Q and design D. Rating R is calculated by the formula
R=4(2F+2Q+D)−0.01P. The table shows the prices and performance of four models of meat grinders.
transcript
1 Selection of a set or combination 1. To service an international seminar, it is necessary to assemble a group of translators. Information about the candidates is presented in the table. Cost of services Interpreters Languages (rubles per day) 1 German, Spanish English, German English English, French French Spanish 4000 Using the table, gather at least one group in which interpreters together speak four foreign languages: English, German, French and Spanish, and the total cost of their services does not exceed rubles per day. In the answer for the assembled group, indicate the numbers of translators without spaces, commas and other additional characters. 2. To serve an international seminar, it is necessary to assemble a group of translators. Information about the candidates is presented in the table. Cost of services Interpreters Languages (rubles per day) 1 German, Spanish English, German English English, French French Spanish 4000 Using the table, gather at least one group in which interpreters together speak four foreign languages: English, German, French and Spanish, and the total cost of their services does not exceed rubles per day. Please provide exactly one set of translator numbers without spaces, commas or other additional characters in your answer. 3. For paid individual lessons with students, it is required to assemble a group of teachers. Information about teachers is presented in the table. Cost of services Teachers Subjects (rubles per course) 1 Computer science, physics Mathematics, physics 9000
2 3 Mathematics, English Computer science Mathematics English 3000 Using the table, gather at least one group in which teachers together can teach in all four subjects, and the total cost of their services does not exceed rubles. 4. A tourist chooses an excursion program for himself. Information about some museums and parks, prepared by the tourist office, is presented in the table. Excursion number Point of interest Opening hours Time (in hours) for travel and visiting , Peterhof 10:00 19:00 6 Using the table, select an excursion program so that the tourist visits at least three attractions in one day. In the answer for the selected program, indicate the numbers of excursions without spaces, commas and other additional characters. 5. Mikhail decided to visit the Amusement Park. Information about tickets for attractions is presented in the table. Some tickets allow you to visit two attractions at once. Ticket number Visited attractions Cost 1 Roller coaster Panic room, roller coaster Autodrom, roller coaster Ferris wheel Ferris wheel, autodrome Autodrome 100 Using the table, select a set of tickets so that Mikhail visits all four attractions: Ferris wheel, fear room, roller coaster, autodrome , and the total cost of tickets did not exceed 800 rubles. In your answer, specify exactly one set of ticket numbers without spaces, commas, or other additional characters.
3 6. A traveler from Moscow wants to visit four cities of the Golden Ring of Russia: Vladimir, Yaroslavl, Suzdal and Rostov. The travel agency offers routes with visits to some cities of the Golden Ring. Information about the cost of tickets and the composition of the routes is presented in the table. Route number Visited cities Cost 1 Suzdal, Yaroslavl, Vladimir Rostov, Vladimir Yaroslavl, Vladimir Suzdal Rostov, Suzdal Yaroslavl, Rostov 2350 Which routes should a traveler choose to visit all four cities and spend less than 5000 rubles on all trips? In your response, specify exactly one set of routes without spaces, commas, or other additional characters. 7. A tourist who has arrived in St. Petersburg wants to visit four museums: the Hermitage, the Russian Museum, the Peter and Paul Fortress and St. Isaac's Cathedral. Excursion ticket offices offer routes with a visit to one or more objects. Information about the cost of tickets and the composition of the routes is presented in the table. Cost Route number Visited objects 1 Hermitage St. Isaac's Cathedral, Peter and Paul Fortress Hermitage, Peter and Paul Fortress Peter and Paul Fortress Russian Museum St. Isaac's Cathedral, Russian Museum 550 Which routes should a tourist choose to visit all four museums and spend the least amount on all tickets? Specify exactly one set of route numbers in your response without spaces, commas, or other additional characters. 8. The tourist selects excursions. Information about excursions is presented in the table. Excursion number Visited objects Cost 1 Fortress, country palace Country palace 50 3 Museum of painting Park Park, museum of painting 300
4 6 Park, fortress 350 Using the table, select a set of excursions so that the tourist visits four objects: a fortress, a country palace, a park and a museum of art, and the total cost of excursions would not exceed 650 rubles. In your answer, specify exactly one set of tour numbers without spaces, commas, or other additional characters. 9. To process a summer cottage, a summer resident needs to purchase a shovel, chopper, pitchfork and rake. The store sells sets of tools, some sets consist of only one tool. Prices are shown in the table. Cost Set Number Tools 1 Shovel, Pitchfork Pitchfork Rake Shovel Chopper, Rake Chopper, Pitchfork 460 Using the table, assemble a complete set of necessary tools so that the total cost is the lowest. In the response for the assembled kit, indicate the kit numbers without spaces, commas, and other additional characters. 10. There are five attractions in the city park: a carousel, a Ferris wheel, an autodrome, Chamomile and a Merry Shooting Range. The box office sells six types of tickets, each of which allows you to visit one or two attractions. Information about the cost of tickets is presented in the table. Ticket type A set of attractions Cost 1 Merry Shooting Range, Chamomile Merry Shooting Range, carousel Autodrome, Ferris wheel Romashka Chamomile, autodrome Ferris wheel, carousel 400 Andrey wants to visit all five attractions, but has only 900 available rubles. What kind of tickets should he buy? In your answer, indicate the numbers corresponding to the types of tickets, without spaces, commas and other additional characters. 11. There are five attractions in the city park: a carousel, a Ferris wheel, an autodrome, Chamomile and a Merry Shooting Range. The box office sells six types of tickets, each of which allows you to visit one or two attractions. Information about the cost of tickets is presented in the table.
5 Ticket number A set of attractions Cost 1 "Romashka" Ferris wheel, carousel Autodrom, Ferris wheel "Romashka", autodrome "Merry Shooting Range", carousel "Merry Shooting Range", "Romashka" 400 Andrey wants to visit all five attractions, but has only 900 rubles. What kind of tickets should he buy? 12. Mikhail decided to visit the Amusement Park. Information about tickets for attractions is presented in the table. Some tickets allow you to visit two attractions at once. Cost Ticket number Attractions 1 Panic room, fun room Autodrome Ferris wheel Fun room Ferris wheel, autodrome Autodrome, fun room 400 Using the table, select tickets so that Mikhail visits all four attractions: the total cost of tickets did not exceed 900 rubles. In your answer, indicate any one set of ticket numbers without spaces, commas, and other additional characters. 13. A traveler from Moscow wants to visit four cities of the Golden Ring of Russia: Vladimir, Yaroslavl, Suzdal and Rostov the Great. The travel agency offers routes with visits to some cities of the Golden Ring. Information about the cost of tickets and routes are presented in the table. Route number Visited cities Cost 1 Suzdal, Yaroslavl, Vladimir Rostov, Vladimir Yaroslavl, Vladimir Suzdal Rostov, Suzdal Yaroslavl, Rostov 2350
6 What routes should a traveler choose to visit all four cities and spend less than 5,000 rubles? Please provide a single set of route numbers without spaces, commas, or other additional characters in your response. 14. A tourist who has arrived in St. Petersburg wants to visit 4 museums: the Hermitage, the Russian Museum, the Peter and Paul Fortress and St. Isaac's Cathedral. Excursion ticket offices offer routes with a visit to one or more objects. Information about the cost of tickets and the composition of the routes is presented in the table. Ticket number Visited objects Cost 1 Hermitage Hermitage, Russian Museum St. Isaac's Cathedral Peter and Paul Fortress, St. Isaac's Cathedral Russian Museum 350 Peter and Paul Fortress, Russian Museum Which routes should a traveler choose to visit all four museums and spend the least amount on all tickets? Please provide a single set of route numbers without spaces, commas, or other additional characters in your response. 15. Mikhail decided to visit an amusement park. Information about tickets for attractions is presented in the table. Some tickets allow you to visit two attractions at once. Cost Ticket number Attractions 1 Panic room, fun room Autodrome Ferris wheel Fun room Ferris wheel, autodrome Autodrome, fun room 400 Using the table, select tickets so that Mikhail visits all four attractions: the total cost of tickets did not exceed 900 rubles. In your answer, indicate any one set of ticket numbers without spaces, commas, and other additional characters.
7 16. To service an international seminar, it is necessary to assemble a group of translators. Information about the candidates is presented in the table. Interpreter number Languages Cost of services (rubles per day) 1 English, Spanish English Spanish, French German English, German French 4050 Using the table, gather at least one group in which interpreters together speak all four languages: English, German, Spanish and French, and the total cost of their services does not exceed rubles per day. In your answer, indicate any one set of translator numbers without spaces, commas, or other additional characters. Choice of two options 1. A family of three is traveling from St. Petersburg to Vologda. You can travel by train or you can drive your own car. A train ticket for one person costs 660 rubles. The car consumes 8 liters of gasoline per 100 kilometers, the distance along the highway is 700 km, and the price of gasoline is 19.5 rubles per liter. How many rubles will you have to pay for the cheapest trip for three? 2. To build a garage, you can use one of two types of foundation: concrete or foam block foundation. For a foam block foundation, 2 cubic meters of foam blocks and 4 bags of cement are required. For a concrete foundation, 2 tons of crushed stone and 20 bags of cement are needed. A cubic meter of foam blocks costs 2,450 rubles, crushed stone costs 620 rubles per ton, and a bag of cement costs 230 rubles. How much will the material cost if you choose the cheapest option? 3. In order to knit a sweater, the hostess needs 400 grams of blue wool. You can buy blue yarn at a price of 60 rubles per 50 g, or you can buy undyed yarn at a price of 50 rubles per 50 g and dye it. One bag of paint costs 10 rubles and is designed for dyeing 200 g of yarn. Which purchase option is cheaper? In response, write how much this purchase will cost. 4. When building a rural house, one of two types of foundation can be used: stone or concrete. For a stone foundation, 9 tons of natural stone and 9 bags of cement are needed. For a concrete foundation, 7 tons of crushed stone and 50 bags of cement are needed. A ton of stone costs roubles, crushed stone costs 780 rubles per ton, and a bag of cement costs 230 rubles. How many rubles will the material for the foundation cost if you choose the cheapest option? 5. On average, citizen A. consumes 120 kWh of electricity per month during the daytime, and 185 kWh of electricity at night. Previously, A. had a
8 single-tariff meter, and he paid for all electricity at a rate of 2.40 rubles. per kWh. A year ago, A. installed a two-tariff meter, while daily electricity consumption is paid at a rate of 2.40 rubles. per kWh, and night consumption is paid at the rate of 0.60 rubles. per kWh. During 12 months, the mode of consumption and tariffs for payment of electricity did not change. How much more would A. have paid during this period if the meter had not changed? Give your answer in rubles. On average, citizen A. consumes 120 kWh of electricity per month during the daytime, and 185 kWh of electricity at night. Previously, A. had a one-rate meter installed in his apartment, and he paid for all electricity at the rate of 2.40 rubles. per kWh. A year ago, A. installed a two-tariff meter, while daily electricity consumption is paid at a rate of 2.40 rubles. per kWh, and night consumption is paid at the rate of 0.60 rubles. per kWh. During 12 months, the mode of consumption and tariffs for payment of electricity did not change. How much more would A. have paid during this period if the meter had not changed? Give your answer in rubles. 6. In order to knit a sweater, the hostess needs 600 grams of red wool yarn. You can buy red yarn at a price of 60 rubles per 50 grams, or you can buy undyed yarn at a price of 50 rubles per 50 grams and dye it. One bag of paint costs 50 rubles and is designed for dyeing 300 grams of yarn. Which purchase option is cheaper? In the answer, write how much this purchase will cost. 7. In order to knit a sweater, the hostess needs 800 grams of red wool yarn. You can buy red yarn at a price of 70 rubles per 50 grams, or you can buy undyed yarn at a price of 50 rubles per 50 grams and dye it. One bag of paint costs 40 rubles and is designed for dyeing 400 grams of yarn. Which purchase option is cheaper? In the answer, write how much this purchase will cost. Choice of three options 1. To transport 45 tons of cargo for 1300 km, you can use the services of one of three carrier companies. The cost of transportation and the carrying capacity of vehicles for each carrier is indicated in the table. How many rubles will you have to pay for the cheapest transportation? Carrier Cost of transportation by one car (rubles per 100 km) A.5 B C Carrying capacity of cars (tons) 2. An Internet provider (a company that provides services for connecting to the Internet) offers three tariff plans. Tariff plan Subscription fee Traffic fee Plan "0" No 2.5 rub. per 1 Mb Plan "500" Plan "800" 550 rub. for 500 Mb of traffic per month 2 rubles. for 1 MB over 500 MB 700 rub. for 800 Mb of traffic per month 1.5 rubles. per 1 MB over 800 MB
9 The user assumes that his traffic will be 600 Mb per month and, based on this, chooses the cheapest tariff plan. How many rubles will a user pay per month if his traffic is really 600 MB? 3. To make bookshelves, you need to order 48 identical glasses from one of three companies. The area of each glass is 0.25. The table shows prices for glass, as well as glass cutting and edge grinding. How much will the cheapest order cost? Company Price of glass (rubles per 1 m 2) Cutting and grinding (rubles per glass) A B C For glazing museum showcases, you need to order 20 identical glasses from one of three companies. The area of each glass is 0.25 m2. The table shows the prices for glass and glass cutting. How much will the cheapest order cost? Company Glass price Glass cutting (rubles per 1 m 2 Additional conditions) (rubles per glass) A B C When ordering more than 2500 rubles. cutting free. 5. The client wants to rent a car for a day for a trip of 500 km. The table shows the characteristics of three cars and the cost of their rental. In addition to the rental, the client is obliged to pay fuel for the car for the entire trip. What amount in rubles will the client pay for rent and fuel if he chooses the cheapest option? A Fuel Fuel consumption (l per 100 km) B Gasoline Rent (rubles per 1 day) C Gas Price of diesel fuel 19 rubles per liter, gasoline 22 rubles per liter, gas 14 rubles per liter. 6. The telephone company provides three service plans to choose from. Tariff plan Subscription fee Time-based 135 rub. per month 0.3 rub. Car Diesel Combined 255 rub. in 450 min. per month Fee for 1 minute of conversation 0.28 rub. for 1 min. over 450 min. per month
10 Unlimited 380 rub. per month The Subscriber chose the cheapest tariff plan, based on the assumption that the total duration of telephone conversations is 650 minutes per month. What amount should he pay per month if the total duration of calls in this month will indeed be 650 minutes? Give your answer in rubles. 7. A construction company needs to purchase 40 cubic meters of building timber from one of three suppliers. What is the lowest cost of such a purchase with delivery (in rubles)? Prices and delivery terms are shown in the table. Supplier Bar price (rubles per 1 m 3) A B C Additional conditions When ordering more than rub. free shipping When ordering more than RUR. free delivery 8. A construction company needs to purchase 75 cubic meters of foam concrete from one of three suppliers. Prices and delivery terms are shown in the table. How many rubles will you have to pay for the cheapest purchase with delivery? Delivery cost Supplier Foam concrete cost (rub. per 1 m 3) Delivery cost A rub. B rub. In rubles Additional conditions When ordering more than RUR. delivery free of charge When ordering more than 80 m 3 delivery is free 9. You can get from home to the dacha by bus, train or fixed-route taxi. The table shows the time that needs to be spent on each section of the path. What is the shortest travel time? Give your answer in hours. By bus By train From home to the bus station 15 min From home to the railway station 25 min. Bus on the way: 2 h 15 min. Train on the way: 1 h 45 min. From the bus stop to the cottage on foot 5 minutes. From the station to the cottage on foot 20 minutes.
11 By fixed-route taxi From the house to the fixed-route taxi stop 25 min. Shuttle taxi on the road: 1 h 35 min. From the minibus stop to the dacha on foot 40 minutes 10. Three roads lead from point A to point D. A truck passes through point B at an average speed of 35 km/h, and a bus passes through point C at an average speed of 30 km/h. The third road has no intermediate points, and a passenger car moves along it at an average speed of 40 km/h. The figure shows the layout of the roads and the distance between points along the roads, expressed in kilometers. All three cars left A at the same time. Which car reached D later than the others? In your answer, indicate how many hours he was on the road. 11. A building contractor plans to buy 5 tons of facing bricks from one of three suppliers. The weight of one brick is 5 kg. Prices and delivery terms are shown in the table. How much will the cheapest purchase option cost? Supplier Brick price (rub. per piece) Delivery cost A No B C Special conditions If the order value is higher than rub. shipping with 50% discount. 12. The table shows the tariffs for the services of three taxi companies. The journey is expected to be 70 minutes long. You need to choose a company in which the order will cost the least. How much will this order cost? Taxi company Delivery of the car Duration and cost of the minimum trip * A 350 rubles. No 13 rub. B Free 20 min. 300 rub. 19 rub. At 180 rubles. 10 minutes. 150 rub. 15 rub. Cost of 1 minute in excess of the duration of the minimum trip 13. The cellular communication company decided to provide its regular customer with one of the discounts to choose from. Either a 25% discount on calls to subscribers of other cellular companies in your region, or a 5% discount on calls to other regions, or 15% on mobile phone services.
12th Internet. The client looked at the printout of his calls and found out that in a month he spent 300 rubles on calls to subscribers of other companies in his region, 200 rubles on calls to other regions and 400 rubles on mobile Internet. The client assumes that next month the costs will be the same, and, based on this, he chooses the most favorable discount for himself. What discount did the customer choose? In response, write down how many rubles this discount will amount to. 14. In the first bank, one pound sterling can be bought for 47.4 rubles. In the second bank 30 pounds for 1446 rubles. In the third bank, 12 pounds cost 561 rubles. What is the smallest amount (in rubles) you have to pay for 10 pounds sterling? 15. A promotion has been announced in a clothing store: if a buyer purchases goods worth more than a ruble, he receives a certificate for 1,000 rubles, which can be exchanged in the same store for any product with a price not exceeding 1,000 rubles. If the buyer participates in the promotion, he loses the right to return the goods to the store. Buyer I. wants to buy a jacket for 9500 rubles, a shirt for 800 rubles. and a tie for 600 rubles. In which case will I. pay the least for the purchase: 1) I. will buy all three goods at once. 2) I. will buy a jacket and a shirt first, and get a tie for a certificate. 3) I. will buy a jacket and tie first, get a shirt for a certificate. In response, write down how many rubles I. will pay for the purchase in this case. 16. A promotion has been announced in a clothing store: if a buyer purchases goods worth more than a ruble, he receives a 10% discount on the next purchase. If the buyer participates in the promotion, he loses the right to return the goods to the store. Buyer B. wants to buy a jacket for 9300 rubles, a shirt for 1800 rubles. and gloves at the price of 1200 rubles. In which case will B. pay the least for the purchase: 1) B. will buy all three goods at once. 2) B. will buy a jacket and a shirt first, and then gloves at a discount. 3) B. will buy a jacket and gloves first, and then a shirt at a discount. In response, write down how many rubles B. will pay for the purchase in this case. 17. The table shows the average prices (in rubles) for some basic foodstuffs in three Russian cities (as of the beginning of 2010). Product name Tver Lipetsk Barnaul Wheat bread (loaf) Milk (1 liter) Potatoes (1 kg) Cheese (1 kg) Meat (beef) (1 kg) Sunflower oil (1 liter) Determine which of these cities will be the cheapest next a set of products: 2 loaves of wheat bread, 3 kg of potatoes, 1.5 kg of beef, 1 liter of sunflower oil. In response, write down the cost of this set of products in this city (in rubles).
13 18. Vasya downloads a 30 MB file to his computer from the Internet in 28 seconds. Petya uploads a 28 MB file in 24 seconds, and Misha uploads a 38 MB file in 32 seconds. How many seconds will it take to download a 665 MB file to the computer with the fastest download speed? 19. Automobile magazine ranks cars based on safety, comfort, functionality, quality and design. Each individual indicator is evaluated on a 5-point scale. The rating is calculated according to the formula The table shows the ratings of each indicator for three car models. Determine the highest rating of the car models presented in the table. Car model Safety Comfort Functionality Quality Design A B C Ceramic tiles of the same brand are available in three different sizes. Tiles are packed in packs. It is required to buy a tile to cover the floor of a square room with a side of 3 m. The dimensions of the tile, the number of tiles in a pack and the cost of a pack are shown in the table Tile size (cm cm) Number of tiles in a pack p. The price of a pack 20 kr. How much will the cheapest purchase option cost? 21. For a group of foreign guests, you need to buy 10 guidebooks. The necessary guides were found in three online stores. Terms of purchase and delivery are given in the table. Online store Price of one Cost Additional conditions of the delivery guide A No B Delivery is free if the order amount exceeds 3000 rubles. Delivery is free if the order amount exceeds 2500 rubles. Determine in which of the stores the total amount of the purchase, including shipping, will be the smallest. In response, write down the smallest amount in rubles. 22. In three mobile phone stores, the same phone is sold on credit under different conditions. Conditions are given in the table.
14 Salon Phone price Down payment (in % of the price) Loan term (months) Epsilon Delta Monthly payment Omicron Determine in which of the salons the purchase will cost the least (taking into account the overpayment). In response, write down this amount in rubles. 23. An automobile magazine rates cars based on safety, comfort, functionality, quality, and design. Each individual indicator is rated by the magazine's readers on a five-point scale. The rating is calculated according to the formula The table shows the ratings of each indicator for three car models. Determine which car has the highest rating. In response, write down the value of this rating. Model Safety Comfort Functionality Quality Car design A B C The table shows the average prices (in rubles) for some basic foodstuffs in three Russian cities (as of the beginning of 2010). Product name Vladivostok Voronezh Omsk Wheat bread (loaf) Milk (1 liter) Potatoes (1 kg) Cheese (1 kg) Meat (beef) (1 kg) Sunflower oil (1 liter) Determine which of these cities will be the cheapest next set of products: 2 kg of potatoes, 1 kg of cheese, 1 liter of sunflower oil. In response, write down the cost of this set of products in this city (in rubles). 25. To transport 42 tons of cargo for 600 km, you can use the services of one of three carrier companies. The cost of transportation and the carrying capacity of vehicles of each carrier are indicated in the table. Carrier
15 cars (rubles per 100 km) of one car (tons) A B.5 C How many rubles will you have to pay for the cheapest transportation? 26. The Internet provider offers three tariff plans. Tariff Subscription fee plan Traffic fee Plan "0" No 1.2 rub. for 1 MB Plan "700" 600 rubles. for 700 MB of traffic per month 0.9 rubles. for 1 MB over 700 MB 820 rub. for 1000 MB of traffic at 0.7 rubles. for 1 MB over 1000 Plan "1000" MB per month The user assumes that his traffic will be 800 MB per month, and based on this he chooses the cheapest tariff plan. How many rubles will a user have to pay per month if his traffic is really 800 MB? 27. An automobile magazine determines the rating of cars based on safety S, comfort C, functionality F, quality Q and design D. Rating R is calculated by the formula The table shows the performance of three car models. Car model Safety Comfort Functionality Quality Design A B C Find the highest car rating from the models presented in the table. Choice of variant from four possible 1. Furniture showroom concludes agreements with furniture manufacturers. The contracts indicate what percentage of the proceeds from the sale of furniture goes to the income of the furniture salon. Manufacturing company Percentage of revenue received by the salon Notes "Alpha" 5%
16 Alfa 3% up to RUB Items priced over Rs. Beta 6% All products Omikron 4% All products The price list shows prices for four sofas. Determine which sofa sale is most profitable for the salon. In response, write down how many rubles will go to the income of the salon from the sale of this sofa. Manufacturer Product Price "Alpha" Sofa "Koala" rub. "Alpha" "Beta" Sofa "Tumbler" rub. Sofa "Winnie the Pooh" rub. "Omicron" Sofa "Oblomov" rub. 2. An independent expert laboratory ranks household appliances based on a value factor equal to 0.01 of the average price, functionality, quality and design. Each of the indicators is evaluated by an integer from 0 to 4. The final rating is calculated by the formula The table shows the average price and estimates of each indicator for several models of electric meat grinders. Determine the highest rating of the models of electric meat grinders presented in the table. Meat grinder model Average price Functionality Quality Design A B C D Every month, an independent agency determines the ratings of news sites based on information content, efficiency and objectivity of publications. Each individual indicator is evaluated by integers from 2 to 2. The final rating is calculated according to the formula The table shows the estimates of each indicator for several news sites. Determine the highest rating of the news sites presented in the table. Write it down as an answer, rounded to the nearest whole number. Website Informativeness Efficiency Objectivity VoKak.ru NashiNovosti.com Bezvrak.ru 2 2 0
17 Zhizni.net The rating agency determines the value for money rating of electric hair dryers. The rating is calculated based on the average price and ratings for functionality, quality, and design. Each individual indicator is evaluated by experts on a five-point scale with integers from 0 to 4. The final rating is calculated according to the formula The table shows the estimates of each indicator for several models of hair dryers. Determine which model has the lowest rating. In response, write down the value of this rating. Hair dryer model Average price Functionality Quality Design A B C D Rating agency determines the value for money rating of microwave ovens. The rating is calculated based on the average price and ratings for functionality, quality, and design. Each individual indicator is evaluated by experts on a 5-point scale with integers from 0 to 4. The final rating is calculated according to the formula The table shows the estimates of each indicator for several models of furnaces. Determine which model has the highest rating. In response, write down the value of this rating. Oven model Average price Functionality Quality Design A B C D An independent expert laboratory ranks household appliances based on average price, as well as ratings of functionality, quality and design. Each individual indicator is evaluated by experts on a 5-point scale with integers from 0 to 4. The final rating is calculated by the formula: The table contains estimates of each indicator for several models of electric meat grinders. Determine which model has the highest rating. In response, write down the value of this rating. Meat grinder model Average price Functionality Quality Design A B C
18 D An independent expert laboratory ranks household appliances based on the average price. as well as performance evaluations. quality and design Each individual indicator is evaluated by experts on a five-point scale with integers from 0 to 4. The final rating is calculated by the formula The table shows the estimates of each indicator for several models of electric meat grinders. Determine which model has the highest rating. In response, write down the value of this rating. Meat grinder model Average price Functionality Quality Design A B C D
B4. 1. To transport 45 tons of cargo over 1300 km, you can use the services of one of three carrier companies. transportation and carrying capacity of vehicles for each carrier is indicated in the table. How much
All prototypes of task B4 (2013) (26672) One of three carriers can be used to transport 45 tons of cargo for 1300 km. Transportation cost and vehicle capacity for each carrier
All B4 prototypes of 2013 1 B4 task prototype (26672) To transport 45 tons of cargo for 1300 km, you can use the services of one of three carriers
All B4 prototypes of 2014 1 B4 task prototype (26672) To transport 45 tons of cargo for 1300 km, you can use the services of one of three carriers.
A.S. Krutitsky and N.S. Krutitsky. Preparation for the exam in mathematics. http://matematikalegko.ru Open bank of USE assignments in mathematics http://mathege.ru Choosing the best option 5383. For transportation
Total: 18 Prototypes B5 1 To transport 45 tons of cargo for 1300 km, you can use the services of one of three carrier companies. Transportation cost and vehicle capacity for each carrier
C (201) Carrier Cost of transportation by one vehicle (rubles per 100 km) A 3200 3.5 B 100 5 C 9500 12 Carrying capacity (tons)
All prototypes of task B5 (26672) One of three carriers can be used to transport 45 tons of cargo for 1300 km. transportation and vehicle capacity for each carrier is indicated in
TASK 11 Choosing the best option Choosing a set or combination 1. To service an international seminar, it is necessary to assemble a group of translators. Information about the candidates is presented in the table.
Card 59. Tables (mathematics grade 6) 1. To serve an international seminar, it is necessary to assemble a group of translators. Information about the candidates is presented in the table. Translators Languages Cost of services
TASK 12 Choosing the best option Choosing a set or combination 1. To service an international seminar, it is necessary to assemble a group of translators. Information about the candidates is presented in the table.
2015 2016 Preparing for the Unified State Examination in Mathematics Prototypes of tasks from the open bank of USE tasks in mathematics (profile level) http://mathege.ru Choosing the optimal option Alexander and Natalia Krutitskikh www.matematikalegko.ru
Choosing the best option. 1. To service an international seminar, it is necessary to assemble a group of translators. Information about the candidates is presented in the table. Translators Languages 1 German, Spanish 7000
Mathematics of the Unified State Examination 2014 (a system of tasks from an open bank of tasks) Tasks B4 Tasks for the optimal choice Materials were prepared by: Koryanov A. G. (Bryansk); e-mail: [email protected] Nadezhkina N.V. (Irkutsk);
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Decision.
Answer: 235 or 124.
Answer: 235|124
Source: Approbation of the basic USE in mathematics, October 13-17: option 120911.
For paid individual lessons with students, it is required to assemble a group of teachers. Information about teachers is presented in the table.
Using the table, gather at least one group in which teachers together can teach in all four subjects, and the total cost of their services does not exceed 18,000 rubles.
Decision.
There is no teacher who knows only physics, so we choose either the first or the second teacher.
In the first case, in order not to violate the conditions, it remains to take only the fifth and sixth teachers. The cost of services will be 18,000 rubles per day.
In the second case, without violating the conditions, you can take only the fourth and sixth teachers. The cost of services will be the same as in the first case.
Thus, a group of teachers that satisfies all the conditions can be assembled from teachers 1, 5 and 6 or from teachers 2, 4 and 6.
Answer: 156 or 246.
Answer: 156|246|165|561|516|615|651|264|462|426|624|642
Source: RESHU USE
To service an international seminar, it is necessary to assemble a group of translators. Information about the candidates is presented in the table.
| translators | Languages | Service cost (rubles per day) |
|---|---|---|
| 1 | English german | 7000 |
| 2 | Deutsch | 3900 |
| 3 | French | 2000 |
| 4 | Spanish | 2900 |
| 5 | Spanish, English | 5850 |
| 6 | Spanish, French | 6100 |
Using the table, gather at least one group in which translators together speak all four languages: English, German, Spanish and French, and the total cost of their services does not exceed 12,000 rubles per day. Please provide exactly one set of translator numbers without spaces, commas or other additional characters in your answer. List in ascending order of numbers.
Decision.
There is no English-only interpreter, so we choose either the first or the fifth interpreter.
In the first case, in order not to violate the conditions, it remains to take only the third and fourth translators. The cost of services will be 11,900 rubles per day.
In the second case, without violating the conditions, you can take only the second and third interpreters. The cost of services will be 11,750 rubles per day.
Thus, a group of translators that satisfies all the conditions can be assembled from translators 1, 3 and 4 or from translators 2, 3 and 5.
Answer: 134 or 235.
Answer: 134|235|523
Source: Approbation of the basic USE in mathematics, October 13-17: version 120913.
To service an international seminar, it is necessary to assemble a group of translators. Information about the candidates is presented in the table.
Using the table, gather at least one group in which translators together speak all four languages: English, German, Spanish and French, and the total cost of their services does not exceed 12,000 rubles per day. Please provide exactly one set of translator numbers without spaces, commas or other additional characters in your answer. List the numbers of translators in ascending order.
Decision.
There is no English-only interpreter, so either a fourth or fifth interpreter should be selected.
In the first case, in order not to violate the conditions, it remains to take only the first and second translators. The cost of services will be 11,950 rubles per day.
In the second case, without violating the conditions, you can take only the second and third interpreters. The cost of services will be the same as in the first case.
Thus, a group of translators that satisfies all the conditions can be assembled from translators 2, 3 and 5 or from translators 1, 2 and 4.
Answer: 235 or 124.
Answer: 235|124
Source: Approbation of the basic USE in mathematics, October 13-17: option 166081.
To service an international seminar, it is necessary to assemble a group of translators. Information about the candidates is presented in the table.
Using the table, gather at least one group in which translators together speak all four languages: English, German, Spanish and French, and the total cost of their services does not exceed 12,000 rubles per day. Please provide exactly one set of translator numbers without spaces, commas or other additional characters in your answer. List in ascending order of numbers.
Decision.
There is no English-only interpreter, so we choose either a fourth or fifth interpreter.
In the first case, in order not to violate the conditions, it remains to take only the first and second translators. The cost of services will be 11950 rubles per day. In the second case, without violating the conditions, you can take only the second and third interpreters. The cost of services will be the same as in the first case. Thus, a group of translators that satisfies all the conditions can be assembled from translators 2, 3 and 5 or from translators 1, 2 and 4.
Answer: 235 or 124.
Answer: 235|124
Source: Copy of Approbation of the Basic Unified State Examination in Mathematics, October 13-17: option 153691.
To service an international seminar, it is necessary to assemble a group of translators. Information about the candidates is presented in the table.
| Number translator | Languages | Service cost (rubles per day) |
|---|---|---|
| 1 | English, Spanish | 5850 |
| 2 | English | 3000 |
| 3 | Spanish, French | 6800 |
| 4 | Deutsch | 2000 |
| 5 | English german | 5950 |
| 6 | French | 4050 |
Using the table, gather at least one group in which interpreters together speak all four languages: English, German, Spanish and French, and the total cost of their services does not exceed 12,000 rubles per day. In your answer, indicate any one set of translator numbers without spaces, commas, or other additional characters. List in ascending order of numbers.
Decision.
Answer: 146 or 234.
Answer: 146|234
| translators | Languages | Service cost (rubles per day) |
|---|---|---|
| 1 | Deutsch | 3900 |
| 2 | Spanish, English | 6050 |
| 3 | Spanish | 3100 |
| 4 | Spanish, French | 6100 |
| 5 | French | 1900 |
| 6 | English german | 6850 |
List in ascending order of numbers.
Decision.
There is no English-only interpreter, so we choose either a second or a sixth interpreter.
In the first case, in order not to violate the conditions, it remains to take only the first and fifth translators. The cost of services will be 11,850 rubles per day. In the second case, without violating the conditions, you can take either only the fourth interpreter (then the cost of their services will be equal to 12,950, which is higher than the declared amount) or the third and fifth interpreter (the cost of services will be 11,850 as in the first case). Thus, a group of translators that satisfies all conditions can be assembled from translators 1, 2 and 5 or from translators 3, 5 and 6.
Answer: 125 or 356.
Answer: 125|356
To service an international seminar, it is necessary to assemble a group of translators. Information about the candidates is presented in the table.
Using the table, gather at least one group in which translators together speak all four languages: English, German, Spanish and French, and the total cost of their services does not exceed 12,000 rubles per day. In your answer, indicate any one set of translator numbers without spaces, commas, or other additional characters. List in ascending order of numbers.
Decision.
There is no only French interpreter, so we choose either the first or fifth interpreter.
In the first case, without violating the conditions, you can take either only the fourth interpreter (then the cost of their services will be 12,800, which is higher than the declared amount) or the second and sixth interpreters (the cost of services will be 11,850). In the second case, in order not to violate the conditions, it remains to take only the second and third translators. The cost of services will be 11,650 rubles per day. Thus, a group of interpreters that satisfies all the conditions can be assembled from interpreters 1, 2 and 6 or from interpreters 2, 3 and 5.
Answer: 126 or 235.
Answer: 126|235
To service an international seminar, it is necessary to assemble a group of translators. Information about the candidates is presented in the table.
Using the table, gather at least one group in which translators together speak all four languages: English, German, Spanish and French, and the total cost of their services does not exceed 12,000 rubles per day. In your answer, indicate any one set of translator numbers without spaces, commas, or other additional characters. List in ascending order of numbers.
Decision.
There is no Spanish-only interpreter, so we choose either a third or a fourth interpreter.
In the first case, in order not to violate the conditions, it remains to take only the second and fifth translators. The cost of services will be 11,900 rubles per day.
In the second case, without violating the conditions, you can take either only the first translator (then the cost of their services will be equal to 12,800, which is higher than the declared amount) or the fifth and sixth translator (the cost of services will be the same as in the first case, 11,900). Thus, a group of interpreters that satisfies all the conditions can be assembled from interpreters 2, 3 and 5 or from interpreters 4, 5 and 6.
Answer: 235 or 456.
Answer: 235|456
Using the table, gather at least one group in which translators together speak all four languages: English, German, Spanish and French, and the total cost of their services does not exceed 12,000 rubles per day. In your answer, indicate any one set of translator numbers without spaces, commas, or other additional characters. List in ascending order of numbers.
Decision.
There is no German-only interpreter, so we choose either a second or a fourth interpreter.
In the first case, in order not to violate the conditions, it remains to take only the third and sixth translators. The cost of services will be 11,850 rubles per day.
In the second case, without violating the conditions, you can take either only the fifth interpreter (then the cost of their services will be 12,650, which is higher than the declared amount) or the first and third interpreters (the cost of services will be 11,500). Thus, a group of interpreters that satisfies all conditions can be assembled from interpreters 2, 3 and 6 or from interpreters 1, 3 and 4.
Using the table, gather at least one group in which translators together speak all four languages: English, German, Spanish and French, and the total cost of their services does not exceed 12,000 rubles per day. Enter the numbers in ascending order.
Decision.
It is necessary that the group has an interpreter with knowledge of Spanish, so there are 2 options - take either the first interpreter or the third. If we take the first translator, then taking translators who separately know German and French (4 and 6), we can use the services of translators who know four languages at once, having spent 11,900 rubles. If we take a third translator, then taking translators who separately know English and German (2 and 4), we can also use the services of translators who know all four languages, having spent 11,800 rubles. In all other cases, the cost of services exceeds 12,000 rubles.
Answer: 146 or 234.
Answer: 146|234
In task No. 12 of the basic level of the USE in mathematics, we need to choose the best option from several proposed ones, having previously performed simple mathematical operations. Again, since this task belongs to the section of real mathematics, there is no theory here, but only approaches to solutions that I outlined below in the analysis of typical options.
Analysis of typical options for tasks No. 12 USE in mathematics of a basic level
Option 12MB1
Using the table, gather at least one group in which translators together speak four languages: English, German, French and Spanish, and the total cost of their services does not exceed 12,000 rubles per day.
In your answer, indicate any one set of translator numbers without spaces, commas, or other additional characters.
Execution algorithm:
- Choose the language that the least number of translators speak.
- Supplement with the missing languages, keeping an eye on the cost of services.
Decision:
Let's choose the language that the least number of translators know.
German is spoken by two translators 1 and 2. Let's take 2, since the cost of his services is lower, and the budget is limited.
We will supplement the missing languages, keeping an eye on the cost of services.
Only Spanish and French remained. Translators 5 and 6 speak Spanish and French. Let's check the total cost of translation services.
6 000 + 2 000 + 4 000 = 12 000.
This group is suitable for the assignment.
Note: The solution provided is just one of many. That is, in fact, the task is solved by selecting groups and checking the total cost.
Option 12MB2
| Bag number | Length (cm) | Height (cm) | Width (cm) | Weight (kg) |
| 1 | 49 | 31 | 25 | 10,7 |
| 2 | 62 | 47 | 20 | 5,9 |
| 3 | 45 | 37 | 18 | 8,7 |
| 4 | 46 | 35 | 15 | 6,4 |
| 5 | 59 | 40 | 18 | 7,5 |
| 6 | 50 | 40 | 20 | 12,4 |
Execution algorithm:
Decision:
In this task, you need to select all bags whose mass is not more than 10 kg and the sum of overall dimensions does not exceed 115 cm. Let's select all bags whose mass is less than 10 kg. You can immediately discard bags numbered 1 and 6, because. their mass is more than 10 kg.
- for the 2nd: 62 + 47 + 20 \u003d 129 cm;
- for the 3rd: 45 + 37 + 18 = 100 cm;
- for the 4th: 46 + 35 + 15 \u003d 96 cm;
- for the 5th: 59 + 40 + 18 = 117 cm.
Under the stated condition, bags numbered 3 and 4 are suitable.
Option 12MB3
The table shows data on six bags.
| Number | Length | Height | Width | Weight |
| bags | (cm) | (cm) | (cm) | (kg) |
| 1 | 60 | 48 | 15 | 7,6 |
| 2 | 58 | 38 | 13 | 9,8 |
| 3 | 55 | 42 | 23 | 6,8 |
| 4 | 65 | 35 | 15 | 11,3 |
| 5 | 57 | 34 | 21 | 9,6 |
| 6 | 62 | 40 | 10 | 10,5 |
According to the rules of the airline, a bag can be taken in hand luggage, the sum of three dimensions (length, height, width) of which should not exceed 115 cm, and the weight should not exceed 10 kg. What bags can be taken in hand luggage according to the rules of this airline?
In your answer, indicate the numbers of the selected bags without spaces, commas and other additional characters.
Execution algorithm:
- Select all bags weighing less than 10 kg.
- Calculate the sum of the three measurements of the remaining bags.
- Choose from the obtained values those that do not exceed 115 cm.
Decision:
In this task, you need to select all bags whose mass is not more than 10 kg and the sum of overall dimensions does not exceed 115 cm. Let's select all bags whose mass is less than 10 kg. You can immediately discard bags numbered 4 and 6, because. their mass is more than 10 kg.
Calculate the sum of the three dimensions of the remaining bags.
- for the 1st bag: 60+48+15 = 123 cm;
- for the 2nd bag: 58 + 38 + 13 = 109 cm;
- for the 3rd bag: 55 + 42 + 23 = 130 cm;
- for the 5th bag: 57 + 34 + 21 = 112 cm.
Let us choose from the obtained values those that do not exceed 115 cm.
Under the stated condition, bags numbered 2 and 5 are suitable.
Option 12MB4
| Shop number | Store rating | Cost of goods (rub.) | Delivery cost (rub.) |
| 1 | 3 | 13 895 | 400 |
| 2 | 5 | 18 490 | 0 |
| 3 | 5 | 13 513 | 0 |
| 4 | 5 | 13 745 | 390 |
| 5 | 4 | 13 411 | 399 |
| 6 | 4 | 17 489 | 0 |
Execution algorithm:
Decision:
2nd store: 18490 rubles;
3rd store: 13513 rubles;
4th store: 13745 + 390 = 14,135 rubles;
5th store: 13411 + 399 = 13,810 rubles;
6th store: 17489 rubles.
From the calculations it can be seen that the store number 3 provides the lowest price.
Option 12MB5
Sergey Petrovich wants to buy a microwave oven of a certain model in an online store. The table shows 6 offers from different online stores.
| Shop number | Store rating | Cost of goods (rub.) | Delivery cost (rub.) |
| 1 | 4 | 13 790 | 500 |
| 2 | 4 | 16 295 | 500 |
| 3 | 5 | 14 411 | 450 |
| 4 | 3 | 14 463 | 500 |
| 5 | 3,5 | 16 249 | 510 |
| 6 | 4 | 16 900 | 650 |
Sergey Petrovich believes that the purchase should be made in a store with a rating of at least 4. Among the stores that meet this condition, select the offer with the lowest purchase price, including delivery. In the answer, write down the number of the selected store.
Execution algorithm:
- Choose options with a rating of at least 4.
- Calculate the total cost of goods and delivery for the remaining online stores.
- Select, based on the results obtained, the product with the lowest cost.
Decision:
Let's calculate the total cost of goods and delivery for the remaining online stores.
1st store: 13,790+500 = 14,290 rubles;
2nd store: 16,295+500 = 16,795 rubles;
3rd store: 14 411 + 450 = 14861 rubles;
6th store: 16,900 + 650 = 17,550 rubles.
We choose, based on the results obtained, the product with the lowest cost.
From the calculations it can be seen that the store number 1 provides the lowest price.
Option 12MB6
The tourist chooses his excursions. Information about excursions is presented in the table.
Using the table, select excursions so that the tourist visits four objects: a fortress, a country palace, a park and a museum of art, and the total cost of excursions does not exceed 650 rubles.
In response, indicate any one set of excursion numbers without spaces, commas and other additional characters.
Let's take the first line - we have a fortress and a country palace and we give 350 rubles for them, we just need to find a park and a museum of art. Let's take the second and third lines - so we will spend 200 and 150 rubles, and a total of 700 rubles - this does not suit us. Note the fourth line - we are offered to visit the park and the museum of art for 300 rubles. Total with the first line - 650 rubles - it suits us!
Option 12MB7
The construction company plans to buy 70 m3 of foam blocks from one of the three suppliers. Prices and delivery terms are shown in the table.
How much should I pay for the cheapest purchase with delivery?
Pretty typical task. In this case, it is necessary to calculate each option and choose the smallest one in terms of cost.
In this case, in the answer you must write the amount, not the number of the option, be careful!
Consider option A:
2600 70 + 10000 = 182000 + 10000 = 192000
Option B:
2800 70 \u003d 196000 - delivery is free, since the order amount is more than 150,000 rubles
2700 70 = 189000 - delivery is paid, since the amount is less than 200000, total 197000
Comparing all the options, we find that the least cost option is option No. 1, that is, A, in which the amount is 192,000.
Answer: 192000.
Option 12MB8
There are five rides in the city park: a carousel, a Ferris wheel, an autodrome, Chamomile and a Merry Shooting Range. The box office sells six types of tickets, each of which allows you to visit one or two attractions. Information about the cost of tickets is presented in the table.
What tickets should Andrey buy to visit all five attractions and spend no more than 750 rubles?
In your answer, please specify any one set of numbers without spaces, commas, or other additional characters.
Execution algorithm
- Since there are 5 rides to visit, you need to select 2 sets of 2 rides and one that contains 1 ride.
- Pairs of rides must be chosen so that their names do not repeat.
- The lower the cost of the set, the more likely it is to meet the total amount of 750 rubles.
Decision:
1 attraction is only contained in the 4th set, so it must be used. It involves visiting the Ferris wheel, and you will have to spend 150 rubles on it.
You cannot use the 2nd and 3rd sets, because amount 400 rub., 150 rub. (for the 4th set) and any other of the remaining ones will be more than 750 rubles. We check:
750–(400+150)=200 rubles, and there is no set with such a cost in the list.
Similarly, we exclude the 6th set, because 750–(350+150)=250 rubles, and this amount is also not enough to purchase one more set of the remaining ones.
Accordingly, two sets are suitable - the 1st and 5th. Selecting them, we get:
150+2 300=750 rub.
Checking content sets:
- 1st set: "Merry shooting range", autodrome
- 4th set: ferris wheel
- 5th set: carousel, "Chamomile"
Those. We see that the attractions are not repeated.
Option 12MB9
The building contractor plans to buy 20 tons of facing bricks from one of three suppliers. One brick weighs 5 kg. The price of a brick and the terms of delivery of the entire purchase are shown in the table.
How much will the cheapest purchase option with delivery cost?
Execution algorithm
- Convert 20 tons to kg.
- The resulting value is divided by 5. We get the number of bricks that need to be bought.
- The resulting quotient is multiplied in turn by the numbers from the 2nd column of the table (“Brick price”). Thus, we get the cost of all bricks from different suppliers.
- For supplier A: the received product (see item 2) is summed up with the data from the 3rd column (“Delivery cost”). We receive the full cost of the purchase, including delivery.
- For suppliers B and C, we take into account the "Special conditions" from the 4th column of the table.
Decision:
20 tons = 20,000 kg.
20,000: 5 = 4,000 (pcs) bricks required to be purchased.
Supplier A. The cost of the bricks is:
4000 49 \u003d 196,000 (rubles).
Total purchase price:
196,000+8000=204,000 (rubles).
Supplier B. Cost of bricks:
4000 55=220 000 (rubles).
The order amount is more than 200,000 rubles. Then, according to the "Special conditions", delivery is free of charge, and the total cost is: 220,000 (rubles).
Supplier B. Cost of bricks:
4000 62 = 248 000 (rubles).
This amount is more than 240,000 rubles, therefore, according to the "Special Conditions", a 50% discount is provided for delivery, and the delivery cost is equal to
6000:100 50=3000 (rubles).
Hence the total cost:
248,000+3000=251,000 (rubles).
Conclusion: the cheapest option will cost 204,000 rubles.
Option 12MB10
In three mobile phone stores, the same smartphone is sold on credit under different conditions. The conditions are given in the table.
Determine in which of the salons the purchase will cost the least (taking into account the overpayment). In response, write down the cost of this purchase in rubles.
Execution algorithm
- To calculate the total cost of the purchase, we find the amount of the down payment and add to it the amount that will have to be paid in the form of all monthly payments.
- Using this formula, we calculate the cost of a smartphone for all 3 salons. Finding the smallest value.
Decision:
We create a formula for calculating the total cost. It should look like this: Smartphone Cost · Downpayment (%) + Loan Term · Payment Amount.
Calculation for the Epsilon salon: 19800 0.1 + 6 3200 = 1980 + 19200 = 21180 (rubles).
Calculation for the Delta salon: 20200 0.1 + 12 1580 = 2020 + 18960 = 20980 (rubles).
Calculation for the Omicron salon: 20800 0.2 + 6 2900 \u003d 4160 + 17400 \u003d 21560 (rubles).
The smallest among the numbers obtained is 20980. This means that the cheapest purchase will cost in the Delta salon.
Answer: 20980.
Option 12MB11
To service an international seminar, it is necessary to assemble a group of translators. Information about the candidates is presented in the table.
Using the table, gather at least one group in which translators together speak all four languages: English, German, Spanish and French, and the total cost of their services does not exceed 12,000 rubles per day. In your answer, indicate any one set of translator numbers without spaces, commas, or other additional characters.
Execution algorithm
- We are considering translators for whom 2 languages are indicated. We conclude that it is not possible to use 2 such translators, since in one case the total cost of their services exceeds the allowable budget, in others the languages are duplicated.
- We take into account that the 5th translator "costs" much cheaper than the others, and therefore it is advisable to hire him.
- We refuse translator No. 4, since in this case the German language is duplicated.
- We choose from translators Nos. 2 and 3 (who speak two languages) the one that “costs” less.
- From the remaining translators Nos. 1 and 6, we choose the one whose language has not yet been involved.
Decision:
We cannot take a couple of translators with two languages, because:
- translators No. 2 and No. 3, as well as No. 2 and No. 4, have duplicated languages. In the 1st case it is English, in the 2nd case it is French;
- for translators No. 3 and No. 4, the total cost of services is more than 12,000 rubles. We check: 6800 + 5850 = 12 650 (rubles).
It is definitely worth hiring translator No. 5 in this situation, since his services are the cheapest (1900 rubles). And then you need to refuse translator No. 4 due to the duplication of his German language.
The second take the translator number 2. He speaks 2 languages and at the same time "costs" less than translator No. 3 (also bilingual).
Of the remaining translators No. 1 and No. 6, we refuse No. 1, because French is already represented by translator No. 2.
So, we get a group of №№2,5,6. We check that the total cost of their services fits into the required 12,000 rubles: 6050 + 1900 + 3900 = 11 850 (rubles)
Option 12MB12
The table shows the results of Olympiads in physics and chemistry in 10 "A" class.
Commendation letters are given to those schoolchildren who have a total score in two Olympiads of more than 130 or at least 70 points in at least one subject.
Indicate the numbers of students 10 "A" who scored less than 70 points in physics and received commendable letters, without spaces, commas and other additional characters.
Execution algorithm
- We analyze the 2nd column of the table (“Score in Physics”), we discard all the numbers of students who received more than 70 points.
- For the rest, we check what score they got in chemistry (in the 3rd column of the table). If the score is 70 or more, then we fix the number of the owner of such a result for the answer. If not, then we sum up the points received and check whether their sum is greater or not. The numbers of students who have received 130 points or more are also recorded for the answer.
Decision:
In the 2nd column of the table there are 4 students who scored less than 70 points in physics. Their numbers are 2, 3, 5 and 7.
The score in chemistry for student No. 2 is 64. This is less than 70, however, having found the sum of 67 (in physics) and 64 (in chemistry), we get 131. From this we conclude: student No. 2 received a diploma.
The score in chemistry for student No. 3 is 36 points. 36<70, поэтому находим сумму баллов:
92<130, а, следовательно, ученик №3 не получил грамоту.
Pupil No. 5 has a chemistry score of 79. This exceeds the limit of 70 points, so student No. 5 received a diploma.
Pupil No. 7 earned 41 points in chemistry. Because this score is less than 70, then you need to look for the sum of the scores in both subjects.
94<130, из чего делаем вывод, что ученик №7 не получил грамоту.
Option 12MB13
The client wants to rent a car for a day for a trip of 500 km. The table shows the characteristics of three cars and the cost of their rental.
In addition to the rental, the client is obliged to pay for fuel for the car for the entire trip. The price of diesel fuel is 25 rubles per liter, gasoline - 35 rubles per liter, gas - 20 rubles per liter. How many rubles will the client pay for rent and fuel if he chooses the cheapest option?
Execution algorithm
- We develop a formula for calculating the costs associated with renting a car for each of the cases. It should multiply the price of fuel by fuel consumption per 100 km, and then by 5 (since fuel consumption is given for every 100 km, and in total you have to drive 500 km). Further, the rent should be added to the resulting work.
- We calculate the costs for cars A, B and C using this formula.
- We compare the results obtained, find the minimum.
Decision:
The calculation formula for determining the cost of a car in general is as follows:
Fuel Price · Fuel Consumption · 5+ Rent.
Here, the “Fuel Price” multiplier means the cost of 1 liter of fuel (data are given after the table). "Fuel Consumption" - data from the 3rd column of the table. "Rent" - data from the 4th column of the table.
car A: 25 7 5+3700=875+3700=4575 (rubles)
car B: 35 10 5+3200=1750+3200=4950 (rubles)
car B: 20 14 5+3200=1400+3200=4600 (rubles)
The smallest among the received amounts - 4575 rubles.
Answer: 4575.
Option 12MB14
Alexey needs a vacuum cleaner. The table shows 6 offers from different stores and their distance from Alexey's house.
Aleksey wants to buy a vacuum cleaner from a shop that is no further than 1.2 km from him. Find the lowest cost of the vacuum cleaner in stores (of the presented ones) that satisfy this condition. Give your answer in rubles.
Execution algorithm
- Using the stipulation in the condition that the distance of the store should be no more than 1.2 km, we discard stores with excessive distance.
- From the remaining stores, we choose the one that offers the lowest price of a vacuum cleaner.
Decision:
We find the numbers of stores that are more than 1.2 km away from Alexei's house. These are Nos. 1, 3, 4, 6. Thus, stores numbered 2 and 5 suit Alexey.
Shop No. 2 offers vacuum cleaners at a price of 4005 rubles, No. 5 - at a price of 3980 rubles. From this we conclude: the lowest cost is 3980 rubles.
Answer: 3980.
Option 12MB15
The ISP offers three tariff plans.
The user assumes that his traffic will be 350 MB per month, and based on this he chooses the cheapest tariff plan. How many rubles will a user have to pay per month if his traffic is really 350 MB?
Execution algorithm
- We discard the "500" plan, since its traffic is not equal to 350 MB.
- We calculate the cost of the traffic fee for the "0" plan. To do this, we multiply 350 MB by 2 rubles.
- We calculate the cost of payment for the plan "100". To do this, to 90 rubles. (for the first 100 MB) we add the cost of the remaining 250 MB required. To calculate the cost of 250 MB, we do the following: multiply 250 by 1.5 (see the condition specified in the 3rd column of the table for the "100" plan).
- We compare the results obtained in paragraphs 1 and 2 with each other. Finding the smallest value.
Decision:
Plan "500" according to the condition of the problem is initially not suitable. Therefore, when searching for the cheapest option, we do not take it into account.
For plan "0" we have: 350 2 = 700 (rubles) - the monthly fee for the user will cost.
For the "100" plan, we get: 90+250 1.5=590 (rubles) - for the whole month, taking into account the monthly fee.
Compare the cost of plans "0" and "100" with each other. We conclude: the cost of the plan "100" is less.
Answer: 590.
Option 12MB16
At the diving competition, the judges gave scores from 0 to 10 to three athletes. The results are shown in the table.
The final score is calculated as follows: the two highest and two lowest marks are discarded, and the remaining three are added together, and their sum is multiplied by the difficulty factor.
In your answer, indicate the numbers of athletes whose final score is more than 140, without spaces, commas and other additional characters.
Execution algorithm
- We calculate the final score for each athlete based on the formula prescribed in the task condition.
- We compare the obtained results with the boundary value of 140.
Decision:
For 1st athlete. 2 highest scores: 8.5 and 7.9. 2 lowest scores: 5.5 and 5.9. Remaining: 6.6; 7.8; 6.6. Final score: (6.6+7.8+6.6) 8=168.
For the 2nd athlete. 2 highest ratings: 8.4 and 8.4. 2 lowest scores: 5.0 and 6.4. Remaining: 7.1; 8.1; 7.6. Final score: (7.1+8.1+7.6) 6=136.8.
For the 3rd athlete. 2 highest ratings: 8.1 and 7.9. 2 lowest scores: 5.4 and 6.4. Remaining: 6.6; 6.5; 7.2. Final score: (6.6+6.5+7.2) 7=142.1.
Compare:
168>140; 136,8<140; 142,1>140.
Option 12MB17
To transport 42 tons of cargo over 1200 km, you can use the services of one of three carrier companies. The cost of transportation and the carrying capacity of vehicles of each carrier are indicated in the table.
How many rubles will you have to pay for the cheapest transportation?
Execution algorithm
- First, we determine the cost of 1 cargo transportation from carrier A. To do this, divide 1200 by 100 and multiply by the number from the 2nd column of the table for carrier A. Then we find the number of cars needed to transport 42 tons of cargo. To do this, we divide 42 by the number from the 3rd column for carrier A. Then we multiply the 2 received numbers.
- We perform similar calculations for carriers B and C.
- We find the minimum among the obtained results.
Decision:
For carrier A. The cost of 1 cargo transportation is:
1200:100 3100=37 200 (rubles).
Cars will be required
42:4=10.5, i.e. 10+1=11.
The total shipping cost is:
37,200 11=409,200 (rubles).
For carrier B. Cost of 1 shipment:
1200:100 4000=48 000 (rubles).
Cars for this will require:
42:5.5≈7.64, i.e. 7+1=8.
Full cost:
48,000 8=384,000 (rubles).
For carrier B. Cost of 1 shipment:
1200:100 7600=91 200 (rubles).
Cars for this you need:
42:10=4,2, i.e. 4+1=5.
Full cost:
91,200 5=456,000 (rubles).
Comparing the results obtained, we see that carrier B offers the lowest total cost.
