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# RD Sharma solutions for Mathematics for Class chapter 30 - Linear programming [Latest edition]

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## Chapter 30: Linear programming

Ex. 30.1Ex. 30.2Ex. 30.3Ex. 30.4Ex. 30.5MCQ

#### RD Sharma solutions for Mathematics for Class Chapter 30 Linear programming Exercise 30.1 [Pages 14 - 17]

Ex. 30.1 | Q 1 | Page 14

A small manufacturing firm produces two types of gadgets A and B, which are first processed in the foundry, then sent to the machine shop for finishing. The number of man-hours of labour required in each shop for the production of each unit of and B, and the number of man-hours the firm has available per week are as follows:

 Gadget Foundry Machine-shop A 10 5 B 6 4 Firm's capacity per week 1000 600

The profit on the sale of A is Rs 30 per unit as compared with Rs 20 per unit of B. The problem is to determine the weekly production of gadgets A and B, so that the total profit is maximized. Formulate this problem as a LPP.

Ex. 30.1 | Q 2 | Page 14

A company is making two products A and B. The cost of producing one unit of products A and B are Rs 60 and Rs 80 respectively. As per the agreement, the company has to supply at least 200 units of product B to its regular customers. One unit of product  requires one machine hour whereas product B has machine hours available abundantly within the company. Total machine hours available for product A are 400 hours. One unit of each product A and B requires one labour hour each and total of 500 labour hours are available. The company wants to minimize the cost of production by satisfying the given requirements. Formulate the problem as a LPP.

Ex. 30.1 | Q 3 | Page 14

A firm manufactures 3 products AB and C. The profits are Rs 3, Rs 2 and Rs 4 respectively. The firm has 2 machines and below is the required processing time in minutes for each machine on each product :

 Machine Products A B C M1M2 4 3 5 2 2 4

Machines M1 and M2 have 2000 and 2500 machine minutes respectively. The firm must manufacture 100 A's, 200 B's and 50 C's but not more than 150 A's. Set up a LPP to maximize the profit.

Ex. 30.1 | Q 4 | Page 15

A firm manufactures two types of products A and B and sells them at a profit of Rs 2 on type A and Rs 3 on type B. Each product is processed on two machines M1 and M2. Type A requires one minute of processing time on M1 and two minutes of M2; type B requires one minute on M1 and one minute on M2. The machine M1 is available for not more than 6 hours 40 minutes while machine M2 is available for 10 hours during any working day. Formulate the problem as a LPP.

Ex. 30.1 | Q 5 | Page 15

A rubber company is engaged in producing three types of tyres AB and C. Each type requires processing in two plants, Plant I and Plant II. The capacities of the two plants, in number of tyres per day, are as follows:

 Plant A B C I 50 100 100 II 60 60 200

The monthly demand for tyre AB and C is 2500, 3000 and 7000 respectively. If plant I costs Rs 2500 per day, and plant II costs Rs 3500 per day to operate, how many days should each be run per month to minimize cost while meeting the demand? Formulate the problem as LPP.

Ex. 30.1 | Q 6 | Page 15

A company sells two different products A and B. The two products are produced in a common production process and are sold in two different markets. The production process has a total capacity of 45000 man-hours. It takes 5 hours to produce a unit of A and 3 hours to produce a unit of B. The market has been surveyed and company officials feel that the maximum number of units of A that can be sold is 7000 and that of B is 10,000. If the profit is Rs 60 per unit for the product A and Rs 40 per unit for the product B, how many units of each product should be sold to maximize profit? Formulate the problem as LPP.

Ex. 30.1 | Q 7 | Page 15

To maintain his health a person must fulfil certain minimum daily requirements for several kinds of nutrients. Assuming that there are only three kinds of nutrients-calcium, protein and calories and the person's diet consists of only two food items, I and II, whose price and nutrient contents are shown in the table below:

 Food I(per lb) Food II(per lb) Minimum daily requirementfor the nutrient Calcium 10 5 20 Protein 5 4 20 Calories 2 6 13 Price (Rs) 60 100

What combination of two food items will satisfy the daily requirement and entail the least cost? Formulate this as a LPP.

Ex. 30.1 | Q 8 | Page 16

A manufacturer can produce two products, A and B, during a given time period. Each of these products requires four different manufacturing operations: grinding, turning, assembling and testing. The manufacturing requirements in hours per unit of products A and B are given below.

 A B Grinding 1 2 Turning 3 1 Assembling 6 3 Testing 5 4

The available capacities of these operations in hours for the given time period are: grinding 30; turning 60, assembling 200; testing 200. The contribution to profit is Rs 20 for each unit of A and Rs 30 for each unit of B. The firm can sell all that it produces at the prevailing market price. Determine the optimum amount of A and B to produce during the given time period. Formulate this as a LPP.

Ex. 30.1 | Q 9 | Page 16

Vitamins A and B are found in two different foods F1 and F2. One unit of food F1contains 2 units of vitamin A and 3 units of vitamin B. One unit of food F2 contains 4 units of vitamin A and 2 units of vitamin B. One unit of food F1 and F2 cost Rs 50 and 25 respectively. The minimum daily requirements for a person of vitamin A and B is 40 and 50 units respectively. Assuming that any thing in excess of daily minimum requirement of vitamin A and B is not harmful, find out the optimum mixture of food F1 and F2 at the minimum cost which meets the daily minimum requirement of vitamin A and B. Formulate this as a LPP.

Ex. 30.1 | Q 10 | Page 16

An automobile manufacturer makes automobiles and trucks in a factory that is divided into two shops. Shop A, which performs the basic assembly operation, must work 5 man-days on each truck but only 2 man-days on each automobile. Shop B, which performs finishing operations, must work 3 man-days for each automobile or truck that it produces. Because of men and machine limitations, shop A has 180 man-days per week available while shop B has 135 man-days per week. If the manufacturer makes a profit of Rs 30000 on each truck and Rs 2000 on each automobile, how many of each should he produce to maximize his profit? Formulate this as a LPP.

Ex. 30.1 | Q 11 | Page 16

A firm manufactures two products, each of which must be processed through two departments, 1 and 2. The hourly requirements per unit for each product in each department, the weekly capacities in each department, selling price per unit, labour cost per unit, and raw material cost per unit are summarized as follows:

 Product A Product B Weekly capacity Department 1 3 2 130 Department 2 4 6 260 Selling price per unit Rs 25 Rs 30 Labour cost per unit Rs 16 Rs 20 Raw material cost per unit Rs 4 Rs 4

The problem is to determine the number of units to produce each product so as to maximize total contribution to profit. Formulate this as a LPP.

Ex. 30.1 | Q 12 | Page 16

An airline agrees to charter planes for a group. The group needs at least 160 first class seats and at least 300 tourist class seats. The airline must use at least two of its model 314 planes which have 20 first class and 30 tourist class seats. The airline will also use some of its model 535 planes which have 20 first class seats and 60 tourist class seats. Each flight of a model 314 plane costs the company Rs 100,000 and each flight of a model 535 plane costs Rs 150,000. How many of each type of plane should be used to minimize the flight cost? Formulate this as a LPP.

Ex. 30.1 | Q 13 | Page 16

Amit's mathematics teacher has given him three very long lists of problems with the instruction to submit not more than 100 of them (correctly solved) for credit. The problem in the first set are worth 5 points each, those in the second set are worth 4 points each, and those in the third set are worth 6 points each. Amit knows from experience that he requires on the average 3 minutes to solve a 5 point problem, 2 minutes to solve a 4 point problem, and 4 minutes to solve a 6 point problem. Because he has other subjects to worry about, he can not afford to devote more than

$3\frac{1}{2}$ hours altogether to his mathematics assignment. Moreover, the first two sets of problems involve numerical calculations and he knows that he cannot stand more than
$2\frac{1}{2}$  hours work on this type of problem. Under these circumstances, how many problems in each of these categories shall he do in order to get maximum possible credit for his efforts? Formulate this as a LPP.

Ex. 30.1 | Q 14 | Page 17

A farmer has a 100 acre farm. He can sell the tomatoes, lettuce, or radishes he can raise. The price he can obtain is Rs 1 per kilogram for tomatoes, Rs 0.75 a head for lettuce and Rs 2 per kilogram for radishes. The average yield per acre is 2000 kgs for radishes, 3000 heads of lettuce and 1000 kilograms of radishes. Fertilizer is available at Rs 0.50 per kg and the amount required per acre is 100 kgs each for tomatoes and lettuce and 50 kilograms for radishes. Labour required for sowing, cultivating and harvesting per acre is 5 man-days for tomatoes and radishes and 6 man-days for lettuce. A total of 400 man-days of labour are available at Rs 20 per man-day. Formulate this problem as a LPP to maximize the farmer's total profit.

Ex. 30.1 | Q 15 | Page 17

A firm has to transport at least 1200 packages daily using large vans which carry 200 packages each and small vans which can take 80 packages each. The cost of engaging each large van is ₹400 and each small van is ₹200. Not more than ₹3000 is to be spent daily on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as a LPP given that the objective is to minimize cost

Ex. 30.1 | Q 16 | Page 17

A firm manufactures two products, each of which must be processed through two departments, 1 and 2. The hourly requirements per unit for each product in each department, the weekly capacities in each department, selling price per unit, labour cost per unit, and raw material cost per unit are summarized as follows:

 Product A Product B Weekly capacity Department 1 3 2 130 Department 2 4 6 260 Selling price per unit ₹ 25 ₹ 30 Labour cost per unit ₹ 16 ₹ 20 Raw material cost per unit ₹ 4 ₹ 4

The problem is to determine the number of units to produce each product so as to maximize total contribution to profit. Formulate this as a LPP.

#### RD Sharma solutions for Mathematics for Class Chapter 30 Linear programming Exercise 30.2 [Pages 32 - 33]

Ex. 30.2 | Q 1 | Page 32

Maximize Z = 5x + 3y
Subject to

$3x + 5y \leq 15$
$5x + 2y \leq 10$
$x, y \geq 0$

Ex. 30.2 | Q 2 | Page 32

Maximize Z = 9x + 3y
Subject to

$2x + 3y \leq 13$

$3x + y \leq 5$

$x, y \geq 0$

Ex. 30.2 | Q 3 | Page 32

Minimize Z = 18x + 10y
Subject to

$4x + y \geq 20$
$2x + 3y \geq 30$
$x, y \geq 0$

Ex. 30.2 | Q 4 | Page 32

Maximize Z = 50x + 30y
Subject to

$2x + y \leq 18$
$3x + 2y \leq 34$
$x, y \geq 0$

Ex. 30.2 | Q 5 | Page 32

Maximize Z = 4x + 3y
subject to

$3x + 4y \leq 24$
$8x + 6y \leq 48$
$x \leq 5$
$y \leq 6$
$x, y \geq 0$

Ex. 30.2 | Q 6 | Page 32

Maximize Z = 15x + 10y
Subject to

$3x + 2y \leq 80$
$2x + 3y \leq 70$
$x, y \geq 0$

Ex. 30.2 | Q 7 | Page 32

Maximize Z = 10x + 6y
Subject to

$3x + y \leq 12$
$2x + 5y \leq 34$
$x, y \geq 0$

Ex. 30.2 | Q 8 | Page 32

Maximize Z = 3x + 4y
Subject to

$2x + 2y \leq 80$
$2x + 4y \leq 120$

Ex. 30.2 | Q 9 | Page 32

Maximize Z = 7x + 10y
Subject to

$x + y \leq 30000$
$y \leq 12000$
$x \geq 6000$
$x \geq y$
$x, y \geq 0$

Ex. 30.2 | Q 10 | Page 32

Minimize Z = 2x + 4y
Subject to

$x + y \geq 8$
$x + 4y \geq 12$
$x \geq 3, y \geq 2$

Ex. 30.2 | Q 11 | Page 32

Minimize Z = 5x + 3y
Subject to

$2x + y \geq 10$
$x + 3y \geq 15$
$x \leq 10$
$y \leq 8$
$x, y \geq 0$

Ex. 30.2 | Q 12 | Page 32

Minimize Z = 30x + 20y
Subject to

$x + y \leq 8$
$x + 4y \geq 12$
$5x + 8y = 20$
$x, y \geq 0$

Ex. 30.2 | Q 13 | Page 32

Maximize Z = 4x + 3y
Subject to

$3x + 4y \leq 24$
$8x + 6y \leq 48$
$x \leq 5$
$y \leq 6$
$x, y \geq 0$

Ex. 30.2 | Q 14 | Page 32

Minimize Z = x − 5y + 20
Subject to

$x - y \geq 0$
$- x + 2y \geq 2$
$x \geq 3$
$y \leq 4$
$x, y \geq 0$

Ex. 30.2 | Q 15 | Page 32

Maximize Z = 3x + 5y
Subject to

$x + 2y \leq 20$
$x + y \leq 15$
$y \leq 5$
$x, y \geq 0$

Ex. 30.2 | Q 16 | Page 32

Minimize Z = 3x1 + 5x2
Subject to

$x_1 + 3 x_2 \geq 3$
$x_1 + x_2 \geq 2$
$x_1 , x_2 \geq 0$

Ex. 30.2 | Q 17 | Page 33

Maximize Z = 2x + 3y
Subject to

$x + y \geq 1$
$10x + y \geq 5$
$x + 10y \geq 1$
$x, y \geq 0$

Ex. 30.2 | Q 18 | Page 33

Maximize Z = −x1 + 2x2
Subject to

$- x_1 + 3 x_2 \leq 10$
$x_1 + x_2 \leq 6$
$x_1 - x_2 \leq 2$
$x_1 , x_2 \geq 0$

Ex. 30.2 | Q 19 | Page 33

Maximize Z = x + y
Subject to

$- 2x + y \leq 1$
$x \leq 2$
$x + y \leq 3$
$x, y \geq 0$

Ex. 30.2 | Q 20 | Page 33

Maximize Z = 3x1 + 4x2, if possible,
Subject to the constraints

$x_1 - x_2 \leq - 1$

$- x_1 + x_2 \leq 0$

$x_1 , x_2 \geq 0$

Ex. 30.2 | Q 21 | Page 33

Maximize Z = 3x + 3y, if possible,
Subject to the constraints

$x - y \leq 1$
$x + y \geq 3$
$x, y \geq 0$

Ex. 30.2 | Q 22 | Page 33

Show the solution zone of the following inequalities on a graph paper:

$5x + y \geq 10$

$x + y \geq 6$

$x + 4y \geq 12$

$x \geq 0, y \geq 0$

Find x and y for which 3x + 2y is minimum subject to these inequalities. Use a graphical method.

Ex. 30.2 | Q 23 | Page 33

Find the maximum and minimum value of 2x + y subject to the constraints:
x + 3y ≥ 6, x − 3y ≤ 3, 3x + 4y ≤ 24, − 3x + 2y ≤ 6, 5x + y ≥ 5, xy ≥ 0.

Ex. 30.2 | Q 24 | Page 33

Find the minimum value of 3x + 5y subject to the constraints
− 2x + y ≤ 4, x + y ≥ 3, x − 2y ≤ 2, xy ≥ 0.

Ex. 30.2 | Q 25 | Page 33

Solved the following linear programming problem graphically:
Maximize Z = 60x + 15y
Subject to constraints

$x + y \leq 50$
$3x + y \leq 90$
$x, y \geq 0$

Ex. 30.2 | Q 26 | Page 33

Find graphically, the maximum value of Z = 2x + 5y, subject to constraints given below:

2x + 4y ≤ 8
3x + y ≤ 6
x + y ≤ 4
x ≥ 0, ≥ 0

Ex. 30.2 | Q 27 | Page 33

Solve the following LPP graphically:
Maximize Z = 20 x + 10 y
Subject to the following constraints

$x +$2$y \leq$28
3x+ $y \leq$24
$x \geq$ 2x.
$y \geq$  0

Ex. 30.2 | Q 28 | Page 33

Solve the following linear programming problem graphically:
Minimize  z = 6 x + 3 y
Subject to the constraints:

4 x + $y \geq$ 80
x + 5 $y \geq$ 115

3 x + 2 $y \leq$ 150
$x \geq$ 0  , $y \geq$ 0

#### RD Sharma solutions for Mathematics for Class Chapter 30 Linear programming Exercise 30.3 [Pages 38 - 40]

Ex. 30.3 | Q 1 | Page 38

A diet of two foods F1 and F2 contains nutrients thiamine, phosphorous and iron. The amount of each nutrient in each of the food (in milligrams per 25 gms) is given in the following table:

 Nutrients Food F1 F2 Thiamine 0.25 0.10 Phosphorous 0.75 1.50 Iron 1.60 0.80

The minimum requirement of the nutrients in the diet are 1.00 mg of thiamine, 7.50 mg of phosphorous and 10.00 mg of iron. The cost of F1 is 20 paise per 25 gms while the cost of F2 is 15 paise per 25 gms. Find the minimum cost of diet.

Ex. 30.3 | Q 2 | Page 38

A diet for a sick person must contain at least 4000 units of vitamins, 50 units of minerals and 1400 of calories. Two foods A and B, are available at a cost of Rs 4 and Rs 3 per unit respectively. If one unit of A contains 200 units of vitamin, 1 unit of mineral and 40 calories and one unit of food B contains 100 units of vitamin, 2 units of minerals and 40 calories, find what combination of foods should be used to have the least cost?

Ex. 30.3 | Q 3 | Page 38

To maintain one's health, a person must fulfil certain minimum daily requirements for the following three nutrients: calcium, protein and calories. The diet consists of only items I and II whose prices and nutrient contents are shown below:

 Food I Food II Minimum daily requirement CalciumProteinCalories 1052 466 202012 Price Rs 0.60 per unit Rs 1.00 per unit

Find the combination of food items so that the cost may be minimum.

Ex. 30.3 | Q 4 | Page 39

A hospital dietician wishes to find the cheapest combination of two foods, A and B, that contains at least 0.5 milligram of thiamin and at least 600 calories. Each unit of Acontains 0.12 milligram of thiamin and 100 calories, while each unit of B contains 0.10 milligram of thiamin and 150 calories. If each food costs 10 paise per unit, how many units of each should be combined at a minimum cost?

Ex. 30.3 | Q 5 | Page 39

A dietician mixes together two kinds of food in such a way that the mixture contains at least 6 units of vitamin A, 7 units of vitamin B, 11 units of vitamin and 9 units of vitamin D. The vitamin contents of 1 kg of food X and 1 kg of food Y are given below:

 VitaminA VitaminB VitaminC VitaminD Food XFood Y 12 11 13 21

One kg food X costs Rs 5, whereas one kg of food Y costs Rs 8. Find the least cost of the mixture which will produce the desired diet.

Ex. 30.3 | Q 6 | Page 39

A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F1and F2 are available. Food F1 costs Rs 4 per unit and F2 costs Rs 6 per unit one unit of food F1 contains 3 units of vitamin A and 4 units of minerals. One unit of food F2contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem and find graphically the minimum cost for diet that consists of mixture of these foods and also meets the mineral nutritional requirements

Ex. 30.3 | Q 7 | Page 39

Kellogg is a new cereal formed of a mixture of bran and rice that contains at least 88 grams of protein and at least 36 milligrams of iron. Knowing that bran contains 80 grams of protein and 40 milligrams of iron per kilogram, and that rice contains 100 grams of protein and 30 milligrams of iron per kilogram, find the minimum cost of producing this new cereal if bran costs Rs 5 per kg and rice costs Rs 4 per kg

Ex. 30.3 | Q 8 | Page 39

A wholesale dealer deals in two kinds, A and B (say) of mixture of nuts. Each kg of mixture A contains 60 grams of almonds, 30 grams of  cashew nuts and 30 grams of hazel nuts. Each kg of mixture B contains 30 grams of almonds, 60 grams of cashew nuts and 180 grams of hazel nuts. The remainder of both mixtures is per nuts. The dealer is contemplating to use mixtures A and B to make a bag which will contain at least 240 grams of almonds, 300 grams of cashew nuts and 540 grams of hazel nuts. Mixture A costs Rs 8 per kg. and mixture B costs Rs 12 per kg. Assuming that mixtures A and B are uniform, use graphical method to determine the number of kg. of each mixture which he should use to minimise the cost of the bag.

Ex. 30.3 | Q 9 | Page 39

One kind of cake requires 300 gm of flour and 15 gm of fat, another kind of cake requires 150 gm of flour and 30 gm of fat. Find the maximum number of cakes which can be made from 7.5 kg of flour and 600 gm of fat, assuming that there is no shortage of the other ingradients used in making the cake. Make it as an LPP and solve it graphically.

Ex. 30.3 | Q 10 | Page 40

Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contains 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.

Ex. 30.3 | Q 11 | Page 40

One kind of cake requires 200 g 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 cakes which can be made from 5 kg of flour and 1 kg of fat assuming that there is no storage of the other ingredients used in making the cakes.

Ex. 30.3 | Q 12 | Page 40

A dietician has to develop a special diet using two foods P and Q. Each packet (containing 30 g) of food P contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. Each packet of the same quantity of food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet requires atleast 240 units of calcium, atleast 460 units of iron and at most 300 units of cholesterol. How many packets of each food should be used to minimise the amount of vitamin A in the diet? What is the minimum of vitamin A.

Ex. 30.3 | Q 13 | Page 40

A farmer mixes two brands P and Q of cattle feed. Brand P, costing ₹250 per bag, contains 2 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing ₹200 per bag contains 1.5 units of nutritional element A, 11.25 units of element B and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?

Ex. 30.3 | Q 14 | Page 40

A dietician wishes to mix together two kinds of food X and Y in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin contents of one kg food is given below:

 Food Vitamin A Vitamin B Vitamin C X 1 2 3 Y 2 2 1

One kg of food X costs ₹16 and one kg of food Y costs ₹20. Find the least cost of the mixture which will produce the required diet?

Ex. 30.3 | Q 15 | Page 40

A fruit grower can use two types of fertilizer in his garden, brand P and Q. The amounts (in kg) of nirogen, phosphoric acid, potash, and chlorine in a bag of each brand are given in the table. Tests indicates that the garden needs at least 240 kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of chlorine.

 kg per bag Brand P Brand P Nitrogen 3 3.5 Phosphoric acid 1 2 Potash 3 1.5 Chlorine 1.5 2

If the grower wants to minimize the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden?

#### RD Sharma solutions for Mathematics for Class Chapter 30 Linear programming Exercise 30.4 [Pages 50 - 57]

Ex. 30.4 | Q 1 | Page 50

If a young man drives his vehicle at 25 km/hr, he has to spend Rs 2 per km on petrol. If he drives it at a faster speed of 40 km/hr, the petrol cost increases to Rs 5/per km. He has Rs 100 to spend on petrol and travel within one hour. Express this as an LPP and solve the same.

Ex. 30.4 | Q 2 | Page 50

A manufacturer has three machines installed in his factory. machines I and II are capable of being operated for at most 12 hours whereas Machine III must operate at least for 5 hours a day. He produces only two items, each requiring the use of three machines. The number of hours required for producing one unit each of the items on the three machines is given in the following table:

 Item Number of hours required by the machine AB I II III 12 21 15/4

He makes a profit of Rs 6.00 on item A and Rs 4.00 on item B. Assuming that he can sell all that he produces, how many of each item should he produces so as to maximize his profit? Determine his maximum profit. Formulate this LPP mathematically and then solve it.

Ex. 30.4 | Q 3 | Page 50

Two tailors, A and B earn Rs 15 and Rs 20 per day respectively. A can stitch 6 shirts and 4 pants  while B can stitch 10 shirts and 4 pants per day. How many days shall each work if it is desired to produce (at least) 60 shirts and 32 pants at a minimum labour cost?

Ex. 30.4 | Q 4 | Page 50

A factory manufactures two types of screws, A and B, each type requiring the use of two machines - an automatic and a hand-operated. It takes 4 minute on the automatic and 6 minutes on the hand-operated machines to manufacture a package of screws 'A', while it takes 6 minutes on the automatic and 3 minutes on the hand-operated machine to manufacture a package of screws 'B'. Each machine is available for at most 4 hours on any day. The manufacturer can sell a package of screws 'A' at a profit of 70 P and screws 'B' at a profit of Rs 1. Assuming that he can sell all the screws he can manufacture, how many packages of each type should the factory owner produce in a day in order to maximize his profit? Determine the maximum profit.

Ex. 30.4 | Q 5 | Page 50

A company produces two types of leather belts, say type A and B. Belt A is a superior quality and belt B is of a lower quality. Profits on each type of belt are Rs 2 and Rs 1.50 per belt, respectively. Each belt of type A requires twice as much time as required by a belt of type B. If all belts were of type B, the company could produce 1000 belts per day. But the supply of leather is sufficient only for 800 belts per day (both A and B combined). Belt A requires a fancy buckle and only 400 fancy buckles are available for this per day. For belt of type B, only 700 buckles are available per day.
How should the company manufacture the two types of belts in order to have a maximum overall profit?

Ex. 30.4 | Q 6 | Page 51

A small manufacturer has employed 5 skilled men and 10 semi-skilled men and makes an article in two qualities deluxe model and an ordinary model. The making of a deluxe model requires 2 hrs. work by a skilled man and 2 hrs. work by a semi-skilled man. The ordinary model requires 1 hr by a skilled man and 3 hrs. by a semi-skilled man. By union rules no man may work more than 8 hrs per day. The manufacturers clear profit on deluxe model is Rs 15 and on an ordinary model is Rs 10. How many of each type should be made in order to maximize his total daily profit.

Ex. 30.4 | Q 7 | Page 51

A manufacturer makes two types A and B of tea-cups. Three machines are needed for the manufacture and the time in minutes required for each cup on the machines is given below:

 Machines I II III AB 126 180 69

Each machine is available for a maximum of 6 hours per day. If the profit on each cup A is 75 paise and that on each cup B is 50 paise, show that 15 tea-cups of type A and 30 of type B should be manufactured in a day to get the maximum profit.

Ex. 30.4 | Q 8 | Page 51

A factory owner purchases two types of machines, A and B, for his factory. The requirements and limitations for the machines are as follows:

 Area occupied by themachine Labour force for eachmachine Daily output inunits Machine AMachine B 1000 sq. m1200 sq. m 12 men8 men 6040

He has an area of 7600 sq. m available and 72 skilled men who can operate the machines.
How many machines of each type should he buy to maximize the daily output?

Ex. 30.4 | Q 9 | Page 51

A company produces two types of goods, A and B, that require gold and silver. Each unit of type A requires 3 gm of silver and 1 gm of gold while that of type B requires 1 gm of silver and 2 gm of gold. The company can produce 9 gm of silver and 8 gm of gold. If each unit of type A brings a profit of Rs 40 and that of type B Rs 50, find the number of units of each type that the company should produce to maximize the profit. What is the maximum profit?

Ex. 30.4 | Q 10 | Page 51

A manufacturer of Furniture makes two products : chairs and tables. processing of these products is done on two machines A and B. A chair requires 2 hrs on machine A and 6 hrs on machine B. A table requires 4 hrs on machine A and 2 hrs on machine B. There are 16 hrs of time per day available on machine A and 30 hrs on machine B. Profit gained by the manufacturer from a chair and a table is Rs 3 and Rs 5 respectively. Find with the help of graph what should be the daily production of each of the two products so as to maximize his profit.

Ex. 30.4 | Q 11 | Page 51

A furniture manufacturing company plans to make two products : chairs and tables. From its available resources which consists of 400 square feet to teak wood and 450 man hours. It is known that to make a chair requires 5 square feet of wood and 10 man-hours and yields a profit of Rs 45, while each table uses 20 square feet of wood and 25 man-hours and yields a profit of Rs 80. How many items of each product should be produced by the company so that the profit is maximum?

Ex. 30.4 | Q 12 | Page 52

A firm manufactures two products A and B. Each product is processed on two machines M1 and M2. Product A requires 4 minutes of processing time on M1 and 8 min. on M2 ; product B requires 4 minutes on M1 and 4 min. on M2. The machine M1 is available for not more than 8 hrs 20 min. while machine M2 is available for 10 hrs. during any working day. The products A and B are sold at a profit of Rs 3 and Rs 4 respectively.
Formulate the problem as a linear programming problem and find how many products of each type should be produced by the firm each day in order to get maximum profit.

Ex. 30.4 | Q 13 | Page 52

A firm manufacturing two types of electric items, A and B, can make a profit of Rs 20 per unit of A and Rs 30 per unit of B. Each unit of A requires 3 motors and 4 transformers and each unit of B requires 2 motors and 4 transformers. The total supply of these per month is restricted to 210 motors and 300 transformers. Type B is an export model requiring a voltage stabilizer which has a supply restricted to 65 units per month. Formulate the linear programing problem for maximum profit and solve it graphically.

Ex. 30.4 | Q 14 | Page 52

A factory uses three different resources for the manufacture of two different products, 20 units of the resources A, 12 units of B and 16 units of C being available. 1 unit of the first product requires 2, 2 and 4 units of the respective resources and 1 unit of the second product requires 4, 2 and 0 units of respective resources. It is known that the first product gives a profit of 2 monetary units per unit and the second 3. Formulate the linear programming problem. How many units of each product should be manufactured for maximizing the profit? Solve it graphically.

Ex. 30.4 | Q 15 | Page 52

A publisher sells a hard cover edition of a text book for Rs 72.00 and paperback edition of the same ext for Rs 40.00. Costs to the publisher are Rs 56.00 and Rs 28.00 per book respectively in addition to weekly costs of Rs 9600.00. Both types require 5 minutes of printing time, although hardcover requires 10 minutes binding time and the paperback requires only 2 minutes. Both the printing and binding operations have 4,800 minutes available each week. How many of each type of book should be produced in order to maximize profit?

Ex. 30.4 | Q 16 | Page 52

A firm manufactures headache pills in two sizes A and B. Size A contains 2 grains of aspirin, 5 grains of bicarbonate and 1 grain of codeine; size B contains 1 grain of aspirin, 8 grains of bicarbonate and 66 grains of codeine. It has been found by users that it requires at least 12 grains of aspirin, 7.4 grains of bicarbonate and 24 grains of codeine for providing immediate effects. Determine graphically the least number of pills a patient should have to get immediate relief. Determine also the quantity of codeine consumed by patient.

Ex. 30.4 | Q 17 | Page 52

A chemical company produces two compounds, A and B. The following table gives the units of ingredients, C and D per kg of compounds A and B as well as minimum requirements of C and D and costs per kg of A and B. Find the quantities of A and B which would give a supply of C and D at a minimum cost.

 Compound Minimum requirement A B Ingredient CIngredient D 13 21 8075 Cost (in Rs) per kg 4 6
Ex. 30.4 | Q 18 | Page 52

A company manufactures two types of novelty Souvenirs made of plywood. Souvenirs of type A require 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 available for assembling. The profit is 50 paise each for type A and 60 paise each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize the profit?

Ex. 30.4 | Q 19 | Page 53

A manufacturer makes two products A and B. Product A sells at Rs 200 each and takes 1/2 hour to make. Product B sells at Rs 300 each and takes 1 hour to make. There is a permanent order for 14 of product A and 16 of product B. A working week consists of 40 hours of production and weekly turnover must not be less than Rs 10000. If the profit on each of product A is Rs 20 and on product B is Rs 30, then how many of each should be produced so that the profit is maximum. Also, find the maximum profit.

Ex. 30.4 | Q 20 | Page 53

A manufacturer produces two types of steel trunks. He has two machines A and B. For completing, the first types of the trunk requires 3 hours on machine A and 3 hours on machine B, whereas the second type of the trunk requires 3 hours on machine A and 2 hours on machine B. Machines A and B can work at most for 18 hours and 15 hours per day respectively. He earns a profit of Rs 30 and Rs 25 per trunk of the first type and the second type respectively. How many trunks of each type must he make each day to make maximum profit?

Ex. 30.4 | Q 21 | Page 53

A manufacturer of patent medicines is preparing a production plan on medicines, A and B. There are sufficient raw materials available to make 20000 bottles of A and 40000 bottles of B, but there are only 45000 bottles into which either of the medicines can be put. Further, it takes 3 hours to prepare enough material to fill 1000 bottles of A, it takes 1 hour to prepare enough material to fill 1000 bottles of B and there are 66 hours available for this operation. The profit is Rs 8 per bottle for A and Rs 7 per bottle for B. How should the manufacturer schedule his production in order to maximize his profit?

Ex. 30.4 | Q 22 | Page 53

An aeroplane can carry a maximum of 200 passengers. A profit of Rs 400 is made on each first class ticket and a profit of Rs 600 is made on each economy class ticket. The airline reserves at least 20 seats of first class. However, at least 4 times as many passengers prefer to travel by economy class to the first class. Determine how many each type of tickets must be sold in order to maximize the profit for the airline. What is the maximum profit.

Ex. 30.4 | Q 23 | Page 53

A gardener has supply of fertilizer of type I which consists of 10% nitrogen and 6% phosphoric acid and type II fertilizer which consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, he finds that he needs at least 14 kg of nitrogen and 14 kg of phosphoric acid for his crop. If the type I fertilizer costs 60 paise per kg and type II fertilizer costs 40 paise per kg, determine how many kilograms of each fertilizer should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?

Ex. 30.4 | Q 24 | Page 53

Anil wants to invest at most Rs 12000 in Saving Certificates and National Saving Bonds. According to rules, he has to invest at least Rs 2000 in Saving Certificates and at least Rs 4000 in National Saving Bonds. If the rate of interest on saving certificate is 8% per annum and the rate of interest on National Saving Bond is 10% per annum, how much money should he invest to earn maximum yearly income? Find also his maximum yearly income.

Ex. 30.4 | Q 25 | Page 53

A man owns a field of area 1000 sq.m. He wants to plant fruit trees in it. He has a sum of Rs 1400 to purchase young trees. He has the choice of two types of trees. Type A requires 10 sq.m of ground per tree and costs Rs 20 per tree and type B requires 20 sq.m of ground per tree and costs Rs 25 per tree. When fully grown, type A produces an average of 20 kg of fruit which can be sold at a profit of Rs 2.00 per kg and type B produces an average of 40 kg of fruit which can be sold at a profit of Rs. 1.50 per kg. How many of each type should be planted to achieve maximum profit when the trees are fully grown? What is the maximum profit?

Ex. 30.4 | Q 26 | Page 53

A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of grinding/cutting machine and sprayer. It takes 2 hours on the grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp while 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 most 20 hours and the grinding/cutting machine for at most 12 hours. The profit from the sale of a lamp is ₹5.00 and a shade is ₹3.00. Assuming that the manufacturer sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximise his profit?

Ex. 30.4 | Q 27 | Page 54

A producer has 30 and 17 units of labour and capital respectively which he can use to produce two type of goods x and y. To produce one unit of x, 2 units of labour and 3 units of capital are required. Similarly, 3 units of labour and 1 unit of capital is required to produce one unit of y. If x and y are priced at Rs 100 and Rs 120 per unit respectively, how should be producer use his resources to maximize the total revenue? Solve the problem graphically.

Ex. 30.4 | Q 28 | Page 54

A firm manufactures two types of products A and B and sells them at a profit of Rs 5 per unit of type A and Rs 3 per unit of type B. Each product is processed on two machines M1 and M2. One unit of type A requires one minute of processing time on M1 and two minutes of processing time on M2, whereas one unit of type B requires one minute of processing time on M1 and one minute on M2. Machines M1 and M2 are respectively available for at most 5 hours and 6 hours in a day. Find out how many units of each type of product should the firm produce a day in order to maximize the profit. Solve the problem graphically.

Ex. 30.4 | Q 29 | Page 54

A small firm manufacturers items A and B. The total number of items A and B that it can manufacture in a day is at the most 24. Item A takes one hour to make while item B takes only half an hour. The maximum time available per day is 16 hours. If the profit on one unit of item A be Rs 300 and one unit of item B be Rs 160, how many of each type of item be produced to maximize the profit? Solve the problem graphically.

Ex. 30.4 | Q 30 | Page 54

A company manufactures two types of toys A and B. Type A requires 5 minutes each for cutting and 10 minutes each for assembling. Type B requires 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours available for cutting and 4 hours available for assembling in a day. The profit is Rs 50 each on type A and Rs 60 each on type B. How many toys of each type should the company manufacture in a day to maximize the profit?

Ex. 30.4 | Q 31 | Page 54

A company manufactures two articles A and B. There are two departments through which these articles are processed: (i) assembly and (ii) finishing departments. The maximum capacity of the first department is 60 hours a week and that of other department is 48 hours per week. The product of each unit of article A requires 4 hours in assembly and 2 hours in finishing and that of each unit of B requires 2 hours in assembly and 4 hours in finishing. If the profit is Rs 6 for each unit of A and Rs 8 for each unit of B, find the number of units of A and B to be produced per week in order to have maximum profit.

Ex. 30.4 | Q 32 | Page 54

A firm makes items A and B and the total number of items it can make in a day is 24. It takes one hour to make an item of A and half an hour to make an item of B. The maximum time available per day is 16 hours. The profit on an item of A is Rs 300 and on one item of B is Rs 160. How many items of each type should be produced to maximize the profit? Solve the problem graphically.

Ex. 30.4 | Q 33 | Page 54

A company sells two different products, A and B. The two products are produced in a common production process, which has a total capacity of 500 man-hours. It takes 5 hours to produce a unit of A and 3 hours to produce a unit of B. The market has been surveyed and company officials feel that the maximum number of unit of A that can be sold is 70 and that for B is 125. If the profit is Rs 20 per unit for the product A and Rs 15 per unit for the product B, how many units of each product should be sold to maximize profit?

Ex. 30.4 | Q 34 | Page 54

A box manufacturer makes large and small boxes from a large piece of cardboard. The large boxes require 4 sq. metre per box while the small boxes require 3 sq. metre per box. The manufacturer is required to make at least three large boxes and at least twice as many small boxes as large boxes. If 60 sq. metre of cardboard is in stock, and if the profits on the large and small boxes are Rs 3 and Rs 2 per box, how many of each should be made in order to maximize the total profit?

Ex. 30.4 | Q 35 | Page 54

A manufacturer makes two products, A and B. Product A sells at Rs 200 each and takes 1/2 hour to make. Product B sells at Rs 300 each and takes 1 hour to make. There is a permanent order for 14 units of product A and 16 units of product B. A working week consists of 40 hours of production and the weekly turn over must not be less than Rs 10000. If the profit on each of product A is Rs 20 and an product B is Rs 30, then how many of each should be produced so that the profit is maximum? Also find the maximum profit.

Ex. 30.4 | Q 36 | Page 55

If a young man drives his vehicle at 25 km/hr, he has to spend ₹2 per km on petrol. If he drives it at a faster speed of 40 km/hr, the petrol cost increases to ₹5 per km. He has ₹100 to spend on petrol and travel within one hour. Express this as an LPP and solve the same.

Ex. 30.4 | Q 37 | Page 55

An oil company has two depots, A and B, with capacities of 7000 litres and 4000 litres respectively. The company is to supply oil to three petrol pumps, DEF whose requirements are 4500, 3000 and 3500 litres respectively. The distance (in km) between the depots and petrol pumps is given in the following table:
Figure
Assuming that the transportation cost per km is Rs 1.00 per litre, how should the delivery be scheduled in order that the transportation cost is minimum?

Ex. 30.4 | Q 38 | Page 55

A small firm manufactures gold rings and chains. The total number of rings and chains manufactured per day is at most 24. It takes 1 hour to make a ring and 30 minutes to make a chain. The maximum number of hours available per day is 16. If the profit on a ring is Rs 300 and that on a chain is Rs 190, find the number of rings and chains that should be manufactured per day, so as to earn the maximum profit. Make it as an LPP and solve it graphically.

Ex. 30.4 | Q 39 | Page 55

A library has to accommodate two different types of books on a shelf. The books are 6 cm and 4 cm thick and weigh 1 kg and  $1\frac{1}{2}$ kg each respectively. The shelf is 96 cm long and atmost can support a weight of 21 kg. How should the shelf be filled with the books of two types in order to include the greatest number of books? Make it as an LPP and solve it graphically.

Ex. 30.4 | Q 40 | Page 55

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. If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find the number of tennis rackets and cricket bats that the factory must manufacture to earn the maximum profit. Make it as an LPP and solve it graphically.

Ex. 30.4 | Q 41 | Page 55

A merchant plans to sell two types of personal computers a desktop model and a portable model that will cost Rs 25,000 and Rs 40,000 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 Rs 70 lakhs and his profit on the desktop model is Rs 4500 and on the portable model is Rs 5000. Make an LPP and solve it graphically.

Ex. 30.4 | Q 42 | Page 55

A cooperative society of farmers has 50 hectares of land to grow two crops X and Y. The profits from crops X and Y per hectare are estimated as ₹10,500 and ₹9,000 respectively. To control weeds, a liquid herbicide has to be used for crops X and Y at the rate of 20 litres and 10 litres per hectare, respectively. Further not more than 800 litres of herbicide should be used in order to protect fish and wildlife using a pond which collects drainage from this land. How much land should be allocated to each crop so as to maximise the total profit of the society?

Ex. 30.4 | Q 43 | Page 56

A manufacturing company makes two models A and B of a product. Each piece of model A requires 9 labour hours for fabricating and 1 labour hour for finishing.  Each piece of model B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available are 180 and 30 respectively. The company makes a profit of ₹8000 on each piece of model A and ₹12000 on each piece of model B. How many pieces of model A and model B should be manufactured per week to realise a maximum profit? What is the maximum profit per week?

Ex. 30.4 | Q 44 | Page 56

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 the maximum profit of the factory when it works at full capacity.

Ex. 30.4 | Q 45 | Page 56

A merchant plans to sell two types of personal computers a desktop model and a portable model that will cost Rs 25,000 and Rs 40,000 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 Rs 70 lakhs and his profit on the desktop model is Rs 4500 and on the portable model is Rs 5000.

Ex. 30.4 | Q 46 | Page 56

A toy company manufactures two types of dolls, A and B. Market tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is at most half of that for dolls of type A. Further, the production level of dolls of type A can exceed three times the production of dolls of other type by at most 600 units. If the company makes profit of ₹ 12 and ₹ 16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximise the profit?

Ex. 30.4 | Q 47 | Page 56

There are two types of fertilizers Fand F2. Fconsists of 10% nitrogen and 6% phosphoric acid and ​Fconsists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds the she needs atleast 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If Fcosts ₹6/kg and Fcosts ₹5/kg, determine how much of each type of fertilizer should be used so that the nutrient requirements are met at minimum cost. What is the minimum cost?

Ex. 30.4 | Q 48 | Page 56

A manufacturer has three machine I, II, III installed in his factory. Machines I and II are capable of being operated for at most 12 hours whereas machine III must be operated for atleast 5 hours a day. She produces only two items M and N each requiring the use of all the three machines.
The number of hours required for producing 1 unit each of M and N on the three machines are given in the following table:

 Items Number of hours required on machines I II III M 1 2 1 N 2 1 1.25

She makes a profit of ₹600 and ₹400 on items M and N respectively. How many of each item should she produce so as to maximise her profit assuming that she can sell all the items that she produced? What will be the maximum profit?

Ex. 30.4 | Q 49 | Page 57

There are two factories located one at place P and the other at place Q. From these locations, a certain commodity is to be delivered to each of the three depots situated at A, B and C. The weekly requirements of the depots are respectively 5, 5 and 4 units of the commodity while the production capacity of the factories at P and Q are respectively 8 and 6 units. The cost of transportation per unit is given below:

 From \ To Cost (in ₹) A B C P 160 100 150 Q 100 120 100

How many units should be transported from each factory to each depot in order that the transportation cost is minimum. What will be the minimum transportation cost?

Ex. 30.4 | Q 50 | Page 57

A manufacturer makes two types of toys A and B. Three machines are needed for this purpose and the time (in minutes) required for each toy on the machines is given below:

 Types of Toys Machines I II III A 12 18 6 B 6 0 9

Each machine is available for a maximum of 6 hours per day. If the profit on each toy of type A is ₹7.50 and that on each toy of type B is ₹5, show that 15 toys of type A and 30 toys of type B should be manufactured in a day to get maximum profit.
Ex. 30.4 | Q 51 | Page 57

An aeroplane can carry a maximum of 200 passengers. A profit of ₹1000 is made on each executive class ticket and a profit of ₹600 is made on each economy class ticket. The airline reserves atleast 20 seats for executive class. However, atleast 4 times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximise the profit of the airline. What is the maximum profit?

Ex. 30.4 | Q 52 | Page 57

A manufacturer considers that men and women workers are equally efficient and so he pays them at the same rate. He has 30 and 17 units of workers (male and female) and capital respectively, which he uses to produce two types of goods A and B. To produce one unit of A, 2 workers and 3 units of capital are required while 3 workers and 1 unit of capital is required to produce one unit of B. If A and B are priced at ₹100 and ₹120 per unit respectively, how should he use his resources to maximise the total revenue? Form the above as an LPP and solve graphically. Do you agree with this view of the manufacturer that men and women workers are equally efficient and so should be paid at the same rate?

Ex. 30.4 | Q 53 | Page 57

A manufacturer produces two products and B. Both the products are processed on two different machines. The available capacity of first machine is 12 hours and that of second machine is 9 hours per day. Each unit of product A requires 3 hours on both machines and each unit of product B requires 2 hours on first machine and 1 hour on second machine. Each unit of product A is sold at ₹7 profit and that of at a profit of ₹4. Find the production level per day for maximum profit graphically.

Ex. 30.4 | Q 54 | Page 57

There are two types of fertilisers 'A' and 'B' . 'A' consists of 12% nitrogen and 5% phosphoric acid whereas 'B' consists of 4% nitrogen and 5% phosphoric acid. After testing the soil conditions, farmer finds that he needs at least 12 kg of nitrogen and 12 kg of phosphoric acid for his crops. If 'A' costs ₹10 per kg and 'B' cost ₹8 per kg, then graphically determine how much of each type of fertiliser should be used so that nutrient requiremnets are met at a minimum cost

Ex. 30.4 | Q 55 | Page 57

A small firm manufactures necklaces and bracelets. The total number of necklaces and bracelets that it can handle per day is at most 24. It takes one hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 16. If the profit on a necklace is Rs 100 and that on a bracelet is Rs 300. Formulate on L.P.P. for finding how many of each should be produced daily to maximize the profit?
It is being given that at least one of each must be produced.

#### RD Sharma solutions for Mathematics for Class Chapter 30 Linear programming Exercise 30.5 [Page 65]

Ex. 30.5 | Q 1 | Page 65

Tow godowns, A and B, have grain storage capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration shops, DE and F, whose requirements are 60, 50 and 40 quintals respectively. The cost of transportation per quintal from the godowns to the shops are given in the following table:

 Transportation cost per quintal(in Rs.) From-> A B To D 6.00 4.00 E 3.00 2.00 F 2.50 3.00

How should the supplies be transported in order that the transportation cost is minimum?

Ex. 30.5 | Q 2 | Page 65

A medical company has factories at two places, A and B. From these places, supply is made to each of its three agencies situated at PQ and R. The monthly requirements of the agencies are respectively 40, 40 and 50 packets of the medicines, while the production capacity of the factories, A and B, are 60 and 70 packets respectively. The transportation cost per packet from the factories to the agencies are given below:

 Transportation Cost per packet(in Rs.) From-> A B To P 5 4 Q 4 2 R 3 5
How many packets from each factory be transported to each agency so that the cost of transportation is minimum? Also find the minimum cost?

#### RD Sharma solutions for Mathematics for Class Chapter 30 Linear programming Exercise MCQ [Pages 67 - 68]

MCQ | Q 1 | Page 67

The solution set of the inequation 2x + y > 5 is

•  half plane that contains the origin

• open half plane not containing the origin

• whole xy-plane except the points lying on the line 2x + y = 5

• none of these

MCQ | Q 2 | Page 67

Objective function of a LPP is

•  a constraint

•  a function to be optimized

•  a relation between the variables

• none of these

MCQ | Q 3 | Page 67

Which of the following sets are convex?

• {(xy) : x2 + y2 ≥ 1}

•  {(xy) : y2 ≥ x}

•  {(xy) : 3x2 + 4y2 ≥ 5}

•  {(xy) : y ≥ 2, y ≤ 4}

MCQ | Q 4 | Page 67

Let X1 and X2 are optimal solutions of a LPP, then

• X = λ X1 + (1 − λ) X2, λ ∈ R is also an optimal solution

• X = λ X1 + (1 − λ) X2, 0 ≤ λ ≤ 1 gives an optimal solution

• X = λ X1 + (1 + λ) X2, 0 ≤ λ ≤ 1 gives an optimal solution

• X = λ X1 + (1 + λ) X2, λ ∈ R gives an optimal solution

MCQ | Q 5 | Page 67

The maximum value of Z = 4x + 2y subjected to the constraints 2x + 3y ≤ 18, x + y ≥ 10 ; xy ≥ 0 is

•  36

• 40

•  20

•  none of these

MCQ | Q 6 | Page 67

The optimal value of the objective function is attained at the points

• given by intersection of inequations with the axes only

•  given by intersection of inequations with x-axis only

•  given by corner points of the feasible region

•  none of these

MCQ | Q 7 | Page 67

The maximum value of Z = 4x + 3y subjected to the constraints 3x + 2y ≥ 160, 5x + 2y ≥ 200, x + 2y ≥ 80; xy ≥ 0 is

• 320

•  300

• 230

•  none of these

MCQ | Q 8 | Page 67

Consider a LPP given by
Minimum Z = 6x + 10y
Subjected to x ≥ 6; y ≥ 2; 2x + y ≥ 10; xy ≥ 0
Redundant constraints in this LPP are

• x ≥ 0, y ≥ 0

• x ≥ 6, 2x + y ≥ 10

•  2x + y ≥ 10

• none of these

MCQ | Q 9 | Page 67

The objective function Z = 4x + 3y can be maximised subjected to the constraints 3x + 4y ≤ 24, 8x + 6y ≤ 48, x ≤ 5, y ≤ 6; xy ≥ 0

•  at only one point

• at two points only

•  at an infinite number of points

•  none of these

MCQ | Q 10 | Page 68

If the constraints in a linear programming problem are changed

• the problem is to be re-evaluated

•  solution is not defined

•  the objective function has to be modified

• the change in constraints is ignored

MCQ | Q 11 | Page 68

Which of the following statements is correct?

• Every LPP admits an optimal solution

•  A LPP admits unique optimal solution

•  If a LPP admits two optimal solution it has an infinite number of optimal solutions

• The set of all feasible solutions of a LPP is not a converse set

MCQ | Q 12 | Page 68

Which of the following is not a convex set?

•  {(xy) : 2x + 5y < 7}

•  {(xy) : x2 + y2 ≤ 4}

•  {x :|x| = 5}

•  {(xy) : 3x2 + 2y2 ≤ 6}

MCQ | Q 13 | Page 68

By graphical method, the solution of linear programming problem

$\text{Maximize}\text{ Z }= 3 x_1 + 5 x_2$
$\text{ Subject } to \text{ 3 } x_1 + 2 x_2 \leq 18$
$x_1 \leq 4$
$x_2 \leq 6$
$x_1 \geq 0, x_2 \geq 0, \text{ is }$
• x1 = 2, x2 = 0, Z = 6

• x1 = 2, x2 = 6, Z = 36

• x1 = 4, x2 = 3, Z = 27

• x1 = 4, x2 = 6, Z = 42

MCQ | Q 14 | Page 68

The region represented by the inequation system xy ≥ 0, y ≤ 6, x + y ≤ 3 is

• unbounded in first and second quadrants

•  none of these

MCQ | Q 15 | Page 68

The point at which the maximum value of x + y, subject to the constraints x + 2y ≤ 70, 2xy ≤ 95, xy ≥ 0 is obtained, is

•  (30, 25)

•  (20, 35)

•  (35, 20)

•  (40, 15)

MCQ | Q 16 | Page 68

The value of objective function is maximum under linear constraints

• at the centre of feasible region

•  at (0, 0)

•  at any vertex of feasible region

•  the vertex which is maximum distance from (0, 0)

MCQ | Q 17 | Page 68

The corner points of the feasible region determined by the following system of linear inequalities:
2x + y ≤ 10, x + 3y ≤ 15, xy ≥ 0 are (0, 0), (5, 0), (3, 4) and (0, 5). Let Z = px + qy, where p, q > 0. Condition on p and q so that the maximum of Z occurs at both (3, 4) and (0, 5) is

• p = q

• p = 2q

• p = 3q

• q = 3p

## Chapter 30: Linear programming

Ex. 30.1Ex. 30.2Ex. 30.3Ex. 30.4Ex. 30.5MCQ

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