Ncert Solutions Maths class 12th
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20. There are two types of fertilizers F1 and F2. F1 consists of 10% nitrogen and 6% phosphoric acid and F2 consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds that she needs at least 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If F1 costs ?. 6/kg and F2 costs ?. 5/kg, determine how much of each type of fertilizer should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?
20. There are two types of fertilizers F1 and F2. F1 consists of 10% nitrogen and 6% phosphoric acid and F2 consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds that she needs at least 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If F1 costs ?. 6/kg and F2 costs ?. 5/kg, determine how much of each type of fertilizer should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?
Let the farmer buy x kg of fertilizer F1 and y kg of fertilizer F2. Therefore,
x ≥ 0 and y ≥ 0
The given information can be complied in a table as follows.
| Nitrogen (%) | Phosphoric Acid (%) | Cost (Rs/kg) |
F1 (x) | 10 | 6 | 6 |
F2 (y) | 5 | 10 | 5 |
Requirement (kg) | 14 | 14 |
F1 consists of 10% nitrogen and F2 consists of 5% nitrogen. However, the farmer requires at least 14 kg of
Upvote 
19. A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F1 and F2 are available. Food F1 costs ?. 4 per unit food and F2 costs ?. 6 per unit. One unit of food F1 contains 3 units of vitamin A and 4 units of minerals. One unit of food F2 contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem. Find the minimum cost for diet that consists of mixture of these two foods and also meets the minimal nutritional requirements.
19. A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F1 and F2 are available. Food F1 costs ?. 4 per unit food and F2 costs ?. 6 per unit. One unit of food F1 contains 3 units of vitamin A and 4 units of minerals. One unit of food F2 contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem. Find the minimum cost for diet that consists of mixture of these two foods and also meets the minimal nutritional requirements.
Let the diet contain x units of food F1 and y units of food F2. Therefore,
x ≥ 0 and y ≥ 0
The given information can be complied in a table as follows.
| Vitamin A (units) | Mineral (units) | Cost per unit (Rs) |
Food F1 (x) | 3 | 4 | 4 |
Food F2 (y) | 6 | 3 | 6 |
Requirement | 80 | 100 |
|
The cost of food F1 is Rs 4 per unit and of Food F2 is ? 6 per unit. Therefore, the constraints are
The mathema
Upvote 18. A merchant plans to sell two types of personal computers – a desktop model and a portable model that will cost Rs. 25000 and Rs. 40000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than ? 70 lakhs and if his profit on the desktop model is ?. 4500 and on portable model is ?. 5000.
18. A merchant plans to sell two types of personal computers – a desktop model and a portable model that will cost Rs. 25000 and Rs. 40000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than ? 70 lakhs and if his profit on the desktop model is ?. 4500 and on portable model is ?. 5000.
Let the merchant stock x desktop models and y portable models. Therefore,
x ≥ 0 and y ≥ 0
The cost of a desktop model is Rs 25000 and of a portable model is Rs 4000. However, the merchant can invest a maximum of Rs 70 lakhs.
The monthly demand of computers will not ex
Upvote 17. A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A requires 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours for assembling. The profit is ?.5 each for type A and ?.6 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize the profit?
17. A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A requires 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours for assembling. The profit is ?.5 each for type A and ?.6 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize the profit?
Let the company manufacture x souvenirs of type A and y souvenirs of type B. Therefore,
x ≥ 0 and y ≥ 0
The given information can be complied in a table as follows.
| Type A | Type B | Availability |
Cutting (min) | 5 | 8 | 3 × 60 + 20 =200 |
Assembling (min) | 10 | 8 | 4 × 60 = 240 |
The profit on type A souvenirs is Rs 5 and on type B souvenirs is Rs 6. Therefore, the constraints are
T
Upvote 16. A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/cutting machine and a sprayer. It takes 2 hours on grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 hours and the grinding/cutting machine for at the most 12 hours. The profit from the sale of a lamp is ?.5 and that from a shade is ?.3. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximize his profit?
16. A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/cutting machine and a sprayer. It takes 2 hours on grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 hours and the grinding/cutting machine for at the most 12 hours. The profit from the sale of a lamp is ?.5 and that from a shade is ?.3. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximize his profit?
Let the cottage industry manufacture x pedestal lamps and y wooden shades. Therefore,
x ≥ 0 and y ≥ 0
The given information can be compiled in a table as follows.
| Lamps | Shades | Availability |
Grinding/Cutting Machine (h) | 2 | 1 | 12 |
Sprayer (h) | 3 | 2 | 20 |
The profit on a lamp is Rs 5 and on the shades is Rs 3. Therefore, the constraints are
The mathematical fo
Upvote 15. A factory manufacturers two types of screws, A and B. Each type of screw requires the use of two machines, an automatic and a hand operated. It takes 4 minutes on the automatic and 6 minutes on hand operated machines to manufacture a package of screw A, while it takes 6 minutes on automatic and 3 minutes on hand operated machines to manufacture a package of screws B. Each machine is available for at the most 4 hours on any day. The manufacturer can sell a package of screws A at a profit of ?. 7 and screws B at a profit of ?. 10. Assuming that he can sell all the screws he manufactures, how many packages of each type should the factory owner produce a day in order to maximize his profit? Determine the maximum profit.
15. A factory manufacturers two types of screws, A and B. Each type of screw requires the use of two machines, an automatic and a hand operated. It takes 4 minutes on the automatic and 6 minutes on hand operated machines to manufacture a package of screw A, while it takes 6 minutes on automatic and 3 minutes on hand operated machines to manufacture a package of screws B. Each machine is available for at the most 4 hours on any day. The manufacturer can sell a package of screws A at a profit of ?. 7 and screws B at a profit of ?. 10. Assuming that he can sell all the screws he manufactures, how many packages of each type should the factory owner produce a day in order to maximize his profit? Determine the maximum profit.
Let the factory manufacture x screws of type A and y screws of type B on each day. Therefore,
x ≥ 0 and y ≥ 0
The given information can be compiled in a table as follows.
| Screw A | Screw B | Availability |
Automatic Machine (min) | 4 | 6 | 4 × 60 =240 |
Hand Operated Machine (min) | 6 | 3 | 4 × 60 =240 |
The profit on a package of screws A is Rs 7 and on the package of screws B is Rs 10. Therefore, t
Upvote 14. A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of ?. 17.50 per package on nuts and ?. 7.00 per package on bolts. How many packages of each should be produced each day so as to maximize his profit, if he operates his machines for at the most 12 hours a day?
14. A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of ?. 17.50 per package on nuts and ?. 7.00 per package on bolts. How many packages of each should be produced each day so as to maximize his profit, if he operates his machines for at the most 12 hours a day?
Let the manufacturer produce x packages of nuts and y packages of bolts. Therefore,
x ≥ 0 and y ≥ 0
The given information can be compiled in a table as follows.
| Nuts | Bolts | Availability |
Machine A (h) | 1 | 3 | 12 |
Machine B (h) | 3 | 1 | 12 |
The profit on a package of nuts is Rs 17.50 and on a package of bolts is Rs 7. Therefore, the constraints a
Upvote 13. A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman’s time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftman’s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftman’s time.
(i) What number of rackets and bats must be made if the factory is to work at full capacity?
(ii) If the profit on a racket and on a bat is Rs. 20 and Rs. 10 respectively, find maximum profit of the factory when it works at full capacity.
13. A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman’s time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftman’s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftman’s time.
(i) What number of rackets and bats must be made if the factory is to work at full capacity?
(ii) If the profit on a racket and on a bat is Rs. 20 and Rs. 10 respectively, find maximum profit of the factory when it works at full capacity.
(i) Let the number of rackets and the number of bats to be made be x and y respectively.
The machine time is not available for more than 42 hours.
The craftsman’s time is not available for more than 24 hours.
The factory is to work at full capacity. Therefore,
On solving these equati
Upvote 12. One kind of cake requires 200g of flour and 25 g of fat and another kind of cake requires 100 g of flour and 50 g of fat. Find the maximum number of cake which can be made from 5 kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes.
12. One kind of cake requires 200g of flour and 25 g of fat and another kind of cake requires 100 g of flour and 50 g of fat. Find the maximum number of cake which can be made from 5 kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes.
Let there be x cakes of first kind and y cakes of second kind. Therefore,
x ≥ 0 and y ≥ 0
The given information can be complied in a table as follows.
| Flour (g) | Fat (g) |
Cakes of first kind, x | 200 | 25 |
Cakes of second kind, y | 100 | 50 |
Availability | 5000 | 1000 |
Total numbers of cakes, Z, that can be made are,
The mathematical formulation of the given problem is
Maximize&nb
Upvote 11. Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs ?. 60/kg and Food Q costs ?. 80/kg. Food P contains 3 units/kg of vitamin A and 5 units/kg of vitamin B while Food Q contains 4 units/kg of vitamin A and 2 units/kg of vitamin B. Determine the minimum cost of the mixture.
11. Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs ?. 60/kg and Food Q costs ?. 80/kg. Food P contains 3 units/kg of vitamin A and 5 units/kg of vitamin B while Food Q contains 4 units/kg of vitamin A and 2 units/kg of vitamin B. Determine the minimum cost of the mixture.
Let the mixture contain x kg of food P and y kg of food Q. Therefore, x ≥ 0 and y ≥ 0
The given information can be compiled in a table as follows.
| Vitamin A (units/kg) | Vitamin B (units/kg) | Cost (Rs/kg) |
Food P | 3 | 5 | 60 |
Food Q | 4 | 2 | 80 |
Requirement (units/kg) | 8 | 11 |
|
The mixture must contain at least 8 units of vitamin A and 11 units of vitamin B. Therefore, the constraints are
Total cost, Z, of purchasing
Upvote