Linear Programming, Inequalities , and Graphing

Question

An apple pie uses 4 cups of apples and 3 cups of flour. An apple cobbler uses 2 cups of apples and 3 cups of flour. You have 16 cups of apples and 15 cups of flour. When you sell these at the farmers market you make $3.00 profit per apple pie and $2.00 profit per apple cobbler. Use linear programming to determine how many apple pies and how many apple cobblers you should make to maximize your profit. let ×= the number of apple pies you make and y= the number of apple cobblers you make. Write an inequality to show the constraint on the amount of apples you have?

Answer 1

We need to write an inequality to show the constraint on the amount of apples we have. We have 16 cups of apples, and each apple pie uses 4 cups of apples, while each apple cobbler uses 2 cups of apples.

4x + 2y ≤ 16

We need to write an inequality to show the constraint on the amount of flour we have. We have 15 cups of flour, and each apple pie uses 3 cups of flour, while each apple cobbler uses 3 cups of flour.

3x + 3y ≤ 15

Both inequalities will have x ≥ 0 and y ≥ 0 since you can't have a negative quantity of a product.

I use Desmos Graphing Calculator to graph the two inequalities to determine the corner points when they are overlapping in the shaded region. The corner points are (0, 0), (0, 5), (4, 0), and (3, 2).

Use the profit equation: z = 3x + 2y, where z is the profit (in dollars)

If we sell no apple pies and no apple cobblers, then z = 3(0) + 2(0) = 0 + 0 = 0

If we sell no apple pies and 5 apple cobblers, then z = 3(0) + 2(5) = 0 + 10 = 10

If we sell 4 apple pies and no apple cobblers, then z = 3(4) + 2(0) = 12 + 0 = 12

If we sell 3 apple pies and 2 apple cobblers, then z = 3(3) + 2(2) = 9 + 4 = 13

You will need to sell 3 apple pies and 2 apple cobblers to maximize your profit, which is $13.