A farmer plans to plant two crops, A and B. The cost of cultivating crop A is $40/acre, whereas that of crop B is $60/acre. The farmer has a maximum of $7,400 available for land cultivation. Each acre of crop A requires 20 labor-hours, and each acre of crop B requires 25 labor-hours. The farmer has a maximum of 3,300 labor-hours available. If he expects to make a profit of $220/acre on crop A and $280/acre on crop B, how many acres of each crop should he plant in order to maximize his profit?
A) 80 acres of crop A, 65 acres of crop B; optimal profit $44,200
B) 0 acres of crop A, 124 acres of crop B; optimal profit $34,720
C) 65 acres of crop A, 80 acres of crop B; optimal profit $36,700
D) 165 acres of crop A, 0 acres of crop B; optimal profit $36,300

Question

Quizplus · Accepted Answer

Let x and y be the number of acres of crop A and crop B respectively. Then we need to maximize the profit function:

Profit = 220x + 280y

subject to the constraints:

40x + 60y ≤ 7400 (maximum budget)
20x + 25y ≤ 3300 (maximum labor-hours)

Simplifying the budget constraint, we have:

2x + 3y ≤ 370  [divide both sides by 20]

Now we can graph the feasible region and find the corner points:

(0, 148)
(60, 52)
(80, 32)
(185, 0)

We can then evaluate the profit function at each corner point:

(0, 148) -> Profit = 41,440
(60, 52) -> Profit = 29,120
(80, 32) -> Profit = 36,700
(185, 0) -> Profit = 40,700

Therefore, the maximum profit of $36,700 is achieved when the farmer plants 65 acres of crop A and 80 acres of crop B.

A Farmer Plans to Plant Two Crops, a and B