Word Problem-college algebra-finding maximum

Question

Food and clothing are shipped to victims of a natural disaster. Each carton of food will feed 14 people, while each carton of clothing will help 6 people. Each 20-cubic-foot box of food weighs 40 pounds and 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 21 000 pounds.
- The total volume must be no more than 7000 cubic feet.

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 -
The number of cartons of clothing is -

 

Can someone help get me on ther ight track to solving this.

Answer 1

Hello Gates! Let's start the problem by defining the variables. Because we are interested to find the number of cartons of food and clothing. Let's set x = cartons of food, and y = cartons of clothing.

 

We know that the total weight has to be less than or equal to 21000 pound. So let's group the information that we know about weights together. You wrote that each box of food weights 40 pounds, so we have 40x (since we set x=cartons of food). And each box of clothing weights 20 pounds, so we have 20y. Therefore, we should have the following first inequality:

40x+20y≤21000

 

We also have a limit on volume, which is no more than 7000 cubic feet. Since each box of food is 20 cubic feet, we have 20x. and we have 5y for clothing. The second inequality should be the following:

20x+5y≤7000

 

Then you solve the inequalities by graphing. After you graph, you are interested in finding the maximum of people you can help. You can help 14 people with each box of food (14x) and 6 people with each box of clothing (6y). What you are going to do is to identify the vertices on of the solution after you graphed the inequalities. I used Desmos to graph my inequalities and the purple area is my solution. The vertices are each corner of the shaded area bounded 

 

After that, you'd want to plug each vertex in to 14x+6y to identify when you will the highest value - helping the most people.