मराठी
महाराष्ट्र राज्य शिक्षण मंडळएचएससी विज्ञान (सामान्य) इयत्ता १२ वी

Find feasible solution for the following system of linear inequation graphically. 3x + 4y ≥ 12, 4x + 7y ≤ 28, x ≥ 0, y ≥ 0 - Mathematics and Statistics

Advertisements
Advertisements

प्रश्न

Find feasible solution for the following system of linear inequation graphically.

3x + 4y ≥ 12, 4x + 7y ≤ 28, x ≥ 0, y ≥ 0

तक्ता
आलेख

उत्तर

Corresponding equation Points on the X-axis Points on the Y-axis Region
3x + 4y = 12 (4, 0) (0, 3) Non-origin side
4x + 7y = 28 (7, 0) (0, 4) origin side

the shaded region is the feasible solution. 

shaalaa.com
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 7: Linear Programming - Miscellaneous exercise 7 [पृष्ठ २४३]

APPEARS IN

बालभारती Mathematics and Statistics 1 (Arts and Science) [English] 12 Standard HSC Maharashtra State Board
पाठ 7 Linear Programming
Miscellaneous exercise 7 | Q II) 4) ii) | पृष्ठ २४३

व्हिडिओ ट्यूटोरियलVIEW ALL [1]

संबंधित प्रश्‍न

Minimize `z=4x+5y ` subject to `2x+y>=7, 2x+3y<=15, x<=3,x>=0, y>=0` solve using graphical method.


A dealer in rural area wishes to purchase a number of sewing machines. He has only Rs 5,760 to invest and has space for at most 20 items for storage. An electronic sewing machine cost him Rs 360 and a manually operated sewing machine Rs 240. He can sell an electronic sewing machine at a profit of Rs 22 and a manually operated sewing machine at a profit of Rs 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Make it as a LPP and solve it graphically.


Solve the following L.P.P graphically:

Maximize: Z = 10x + 25y
Subject to: x ≤ 3, y ≤ 3, x + y ≤ 5, x ≥ 0, y ≥ 0


A manufacturer produces two products A 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 Rs 7 profit and  B at a profit of Rs 4. Find the production level per day for maximum profit graphically.


A company manufactures bicycles and tricycles each of which must be processed through machines A and B. Machine A has maximum of 120 hours available and machine B has maximum of 180 hours available. Manufacturing a bicycle requires 6 hours on machine A and 3 hours on machine B. Manufacturing a tricycle requires 4 hours on machine A and 10 hours on machine B.
If profits are Rs. 180 for a bicycle and Rs. 220 for a tricycle, formulate and solve the L.P.P. to determine the number of bicycles and tricycles that should be manufactured in order to maximize the profit.


Solve the following linear programming problem graphically :

Maximise Z = 7x + 10y subject to the constraints

4x + 6y ≤ 240

6x + 3y ≤ 240

x ≥ 10

x ≥ 0, y ≥ 0


Solve the following L.P.P. graphically: 

Minimise Z = 5x + 10y

Subject to x + 2y ≤ 120

Constraints x + y ≥ 60

x – 2y ≥ 0 and x, y ≥ 0


Solve the following L.P.P graphically: Maximise Z = 20x + 10y

Subject to the following constraints x + 2y ≤ 28,

3x + y ≤ 24,

x ≥ 2,

 x, y ≥ 0


Solve the following LPP by graphical method:

Minimize Z = 7x + y subject to 5x + y ≥ 5, x + y ≥ 3, x ≥ 0, y ≥ 0


In order to supplement daily diet, a person wishes to take X and Y tablets. The contents (in milligrams per tablet) of iron, calcium and vitamins in X and Y are given as below :

Tablets  Iron Calcium Vitamin
x 6 3 2
y 2 3 4

The person needs to supplement at least 18 milligrams of iron, 21 milligrams of calcium and 16 milligrams of vitamins. The price of each tablet of X and Y is Rs 2 and Rs 1 respectively. How many tablets of each type should the person take in order to satisfy the above requirement at the minimum cost? Make an LPP and solve graphically.


Maximize Z = 5x + 3y
Subject to

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


Minimize Z = 18x + 10y
Subject to 

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


Maximize Z = 4x + 3y
subject to

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


Maximize Z = 15x + 10y
Subject to 

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

 


Maximize Z = 10x + 6y
Subject to

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


Minimize Z = 30x + 20y
Subject to 

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


Maximize Z = 4x + 3y
Subject to 

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


Minimize Z = 3x1 + 5x2
Subject to

\[x_1 + 3 x_2 \geq 3\]
\[ x_1 + x_2 \geq 2\]
\[ x_1 , x_2 \geq 0\]

 


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


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

\[x + y \leq 50\]
\[3x + y \leq 90\]
\[ x, y \geq 0\]


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   


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.


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?


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


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?


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 the
machine
Labour force for each
machine
Daily output in
units
Machine A
Machine B
1000 sq. m
1200 sq. m
12 men
8 men
60
40

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?


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?


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.


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 C
Ingredient D
1
3
2
1
80
75
Cost (in Rs) per kg 4 6 -

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.


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?


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.


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?


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?    


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.

 


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.


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. 


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?


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?


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


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 and 20 minutes available  for cutting and 4 hours available for assembling. The profit is Rs. 50 each for type A and Rs. 60 each  for type B souvenirs. How many souvenirs of each type should the company manufacture in order to  maximize profit? Formulate the above LPP and solve it graphically and also find the maximum profit. 


A farmer has a supply of chemical fertilizer of type A which contains 10% nitrogen and 6% phosphoric acid and of type B which contains 5% nitrogen and 10% phosphoric acid. After the soil test, it is found that at least 7 kg of nitrogen and the same quantity of phosphoric acid is required for a good crop. The fertilizer of type A costs ₹ 5.00 per kg and the type B costs ₹ 8.00 per kg. Using Linear programming, find how many kilograms of each type of fertilizer should be bought to meet the requirement and for the cost to be minimum. Find the feasible region in the graph.


Find the graphical solution for the system of linear inequation 2x + y ≤ 2, x − y ≤ 1


Find the solution set of inequalities 0 ≤ x ≤ 5, 0 ≤ 2y ≤ 7


Draw the graph of inequalities x ≤ 6, y −2 ≤ 0, x ≥ 0, y ≥ 0 and indicate the feasible region


Area of the region bounded by y = cos x, x = 0, x = π and X-axis is ______ sq.units.


The maximum value of z = 3x + 10y subjected to the conditions 5x + 2y ≤ 10, 3x + 5y ≤ 15, x, y ≥ 0 is ______.


If 4x + 5y ≤ 20, x + y ≥ 3, x ≥ 0, y ≥ 0, maximum 2x + 3y is ______.


Maximise and Minimise Z = 3x – 4y subject to x – 2y ≤ 0, – 3x + y ≤ 4, x – y ≤ 6, x, y ≥ 0


Corner points of the feasible region determined by the system of linear constraints are (0, 3), (1, 1) and (3, 0). Let Z = px + qy, where p, q > 0. Condition on p and q so that the minimum of Z occurs at (3, 0) and (1, 1) is ______.


Let R be the feasible region for a linear programming problem, and let Z = ax + by be the objective function. If R is bounded, then the objective function Z has both a maximum and a minimum value on R and ____________.


In Corner point method for solving a linear programming problem the first step is to ____________.


In the Corner point method for solving a linear programming problem the second step after finding the feasible region of the linear programming problem and determining its corner points is ____________.


The feasible region (shaded) for a L.P.P is shown in the figure. The maximum Z = 5x + 7y is ____________.


Which of the statements describe the solution set for `-2(x + 8) = - 2x + 20`?


Minimise z = – 3x + 4y subject to x + 2y ≤ 8, 3x + 2y ≤ 12, x ≥ 0, y ≥ 0 What will be the minimum value of z ?


The solution set of the inequality 3x + 5y < 4 is ______.


The corner points of the shaded unbounded feasible region of an LPP are (0, 4), (0.6, 1.6) and (3, 0) as shown in the figure. The minimum value of the objective function Z = 4x + 6y occurs at ______.


Solve the following Linear Programming Problem graphically:

Maximize Z = 400x + 300y subject to x + y ≤ 200, x ≤ 40, x ≥ 20, y ≥ 0


The maximum value of z = 5x + 2y, subject to the constraints x + y ≤ 7, x + 2y ≤ 10, x, y ≥ 0 is ______.


The objective function Z = x1 + x2, subject to the constraints are x1 + x2 ≤ 10, – 2x1 + 3x2 ≤ 15, x1 ≤ 6, x1, x2 ≥ 0, has maximum value ______ of the feasible region.


The corner points of the feasible region of a linear programming problem are (0, 4), (8, 0) and `(20/3, 4/3)`. If Z = 30x + 24y is the objective function, then (maximum value of Z – minimum value of Z) is equal to ______.


Solve the following linear programming problem graphically:

Minimize: Z = 5x + 10y

Subject to constraints:

x + 2y ≤ 120, x + y ≥ 60, x – 2y ≥ 0, x ≥ 0, y ≥ 0.


Solve the following linear programming problem graphically:

Maximize: Z = x + 2y

Subject to constraints:

x + 2y ≥ 100,

2x – y ≤ 0

2x + y ≤ 200,

x ≥ 0, y ≥ 0.


Solve the following Linear Programming problem graphically:

Maximize: Z = 3x + 3.5y

Subject to constraints:

x + 2y ≥ 240,

3x + 1.5y ≥ 270,

1.5x + 2y ≤ 310,

x ≥ 0, y ≥ 0.


Solve the following Linear Programming Problem graphically:

Maximize: P = 70x + 40y

Subject to: 3x + 2y ≤ 9,

3x + y ≤ 9,

x ≥ 0,y ≥ 0.


Solve the following Linear Programming Problem graphically:

Minimize: Z = 60x + 80y

Subject to constraints:

3x + 4y ≥ 8

5x + 2y ≥ 11

x, y ≥ 0


Solve the following Linear Programming Problem graphically:

Maximize: z = – x + 2y,

Subject to the constraints: x ≥ 3, x + y ≥ 5, x + 2y ≥ 6, y ≥ 0.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×