Question

Food and clothing are shipped to victims of a natural disaster. Each carton of food will feed 11 people, while each carton of clothing will help 4 people. Each 20-cubic foot box of food weighs 40 pounds each 5-cubic-foot box of clothing weighs 20 pounds. The commercial carriers transporting food and clothing are bound by the following constraints:

- The total weight per carrier cannot exceed 23,000 pounds.

- The total volume must be no more than 9000 cubic feet.

Use this information to answer the following questions.

How many cartons of food and clothing should be sent with each plane shipment to maximize the number of people who can be helped?

The number of cartons of food is ___ cartons.

The number of cartons of clothing is ___ cartons.

Food and clothing are shipped to victims of a natural disaster. Each carton of food will feed 11 people, while each carton of clothing will help 4 people. E the following constraints: The total weight per carrier cannot exceed 23,000 pounds. - The total volume must be no more than 9000 cubic feet. Use this information to answer the following questions How many cartons of food and clothing should be sent with each plano shipment to maximize the number of people who can be helped? The number of cartons of food is cartons. The number of cartons of clothing is cartons.

Answer 1

Answer:

Data given:

Each carton of food will feed 11 people, while each carton of clothing will help 4 people.

Each 20-cubic-foot box of food weighs 40 pounds and each 5-cubic-foot box of clothing weighs 20 pounds.

Constraints on commercial carriers transporting the required cartons -

1. The total weight per carrier cannot exceed 23000 pounds.

2. The total volume must be no more than 9000 cubic feet.

The requirement is to find out the number of cartons of food and clothing that should be sent with each plane shipment to maximise the number of people who can be helped.

Based on the above information, we can convert the above data into a linear programming problem as -

Let "x" represent the number of cartons of food and "y" represent the number of cartons of clothing to be sent on each plane.

Objective function (Maximum help) : M = 11x + 4y (Maximise)

subjected to constraints

Weight constraint - $40x+20y \leq 23000$

Volume constraint -

20.c + 5y < 9000

Now, in order to maximise the objective function, we need to solve the above inequalities graphically and find the feasible region and the corner points.

We can graph the above inequalities separately and then plot them on the same graph as -

1. $40x+20y \leq 23000$

1400 1300 1200 1100 1000- 9000 800- | 700 -600. 500 400 3000 2000 100 -200 -150 -100 -50 0 50 100 150 2000 250 300 350 400 450 500 550 600 650 700 750 -100- -2004

2.   

20.c + 5y < 9000

1400 1300 12000 1100 1000 9000 800- 700 -600. 500 400 3000 2000 100 -200 -150 -100 -50 0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 -100- -2004

Now, we can plot all the inequalities on the same graph as -

1400- 1300 12001 (0, 1150) 1100 1000- 9004 800 |||||| 700- -600. (325, 500) 500 400 300- 200 100 (0.0) (450, 0) -200 -150 -100 -50 0 50 100 150 200 250 300 350 400 450 500 550 - 600 650 700 750 -100 -2004

As we can see from the above graph that the corner points of the feasible region are -  

Therefore, we should evaluate the objective function at all these points as -

a.

.

b.

.

c.

.

d.

.

As we can see that the objective function M is maximum when x = 325 and y = 500.

Hence, the number of cartons of food and clothing that should be sent with each plane shipment to maximise the number of people who can be helped, is 325 and 500 respectively.