I'm having a difficult time understanding this problem. It's a linear programming word problem. Can anyone help me out? You work for an edible arrangements store and are putting together two types...

I'm having a difficult time understanding this problem. It's a linear programming word problem. Can anyone help me out?

You work for an edible arrangements store and are putting together two types of arrangements for graduation parties: regular and special. Regular arrangements require 8 strawberries, 14 grapes, and 7 pineapple chunks. Special arrangements require 12 strawberries, 35 grapes, and 35 pineapple chunks. You have 144 strawberries, 280 grapes, and 35 pineapple chunks available to make arrangements. If you make a profit of $3 each regular and $5 for each special, how many of each should you make in order to maximize your profit?

1. Write an objective function to represent your profit.
2. Write a system of inequalities to represent the constraints.
3. Graph the feasible region.
4. Identify the vertices of your feasible region.
5. Plug in and solver for each of the vertices in your objective function.
6. How many of each arrangement should you make in order to maximize your profit?
7. What is your profit?
8. If the maximum profit is achieved will there be any fruit left over?
9. Explain your reasoning for your answer in number 8

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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In the store there are 144 strawberries, 280 grapes, and 35 pineapple chunks available to make arrangements. A regular arrangement can be made with 8 strawberries, 14 grapes, and 7 pineapple chunks and the profit made when this is sold is $3. A special arrangement with 12 strawberries, 35 grapes, and 35 pineapple chunks can be sold for a profit of $5.

To maximize profits let the number of regular arrangements that are sold be represented by x and the number of special arrangements that are sold be represented by y.

The total profit made is P = 3*x + 5*y

Making the arrangements requires 8*x + 12*y strawberries. As there are 144 strawberries in stock

`8*x + 12*y <= 144`

=> `2*x + 3*y <= 36`

The number of grapes required is 14*x + 35*y and this is less than or equal to 280.

`14*x + 35*y <= 280`

=> `2*x + 5*y <= 40`

The number of pineapple chunks required is 7*x + 35*y. As there are 35 chunks in stock

`7*x + 35*y <= 35`

=> `x + 5*y <= 5`

The feasible regions are represented in the following graph:

As there are only 35 pineapple chunks, it is possible to make only one special arrangement with no regular arrangements. Doing this gives a total profit of $5.

If no special arrangements are made, the maximum number of regular arrangements that can be made is 5.

This gives a total profit of 3*5 = $15

To maximize profits 5 regular arrangements should be made and 0 special arrangements.

Making 5 regular arrangements only uses 40 strawberries and 70 grapes. The rest are left over as the number of pineapple chunks of which there are only 35 available limits the total number of arrangements that can be made.

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