Question

The number of acres a farmer uses for planting pumpkins will be at least 2 times the number of acres for planting corn. The difference between the acres of pumpkin and corn crops will not exceed 10. He will plant between 12 and 18 acres of pumpkins. The profit for each acre of corn is $225 and the profit for each acre of pumpkins is $360.

A) Write the constraints for the situation. Let x be the number of acres of corn and let y be the number of acres of pumpkins.

B) Write the objective function for the situation.

C) Graph the feasible region. Label the vertex points with their coordinates.

D) How many acres of each crop should the farmer plant to maximize the profit? How much is that profit?

Answer 1

Explanation:

A) Let x represent acres of pumpkins, and y represent acres of corn. Here are the constraints:

x ≥ 2y . . . . . pumpkin acres are at least twice corn acres

x - y ≤ 10 . . . . the difference in acreage will not exceed 10

12 ≤ x ≤ 18 . . . . pumpkin acres will be between 12 and 18

0 ≤ y . . . . . the number of corn acres is non-negative

__

B) If we assume the objective is to maximize profit, the profit function we want to maximize is ...

P = 360x +225y

__

C) see below for a graph

__

D) The profit for an acre of pumpkins is the highest, so the farmer should maximize that acreage. The constraint on the number of acres of pumpkins comes from the requirement that it not exceed 18 acres. Then additional profit is maximized by maximizing acres of corn, which can be at most half the number of acres of pumpkins, hence 9 acres.

So profit is maximized for 18 acres of pumpkins and 9 acres of corn.

Maximum profit is $360·18 +$225·9 = $8505.