Chapter 17, Problem 97RE

Profit The total profit from 1 acre of a certain crop depends on the amount spent on fertilizer, x, and on hybrid seed, y, according to the model

P(x, y) = 0.01 (–x² + 3xy + 160x – 5y² + 200y + 2600).

The budget for fertilizer and seed is limited to $280.

(a)    Use the budget constraint to express one variable in terms of the other. Then substitute into the profit function to get a function with one independent variable. Use the method shown in Chapter 14 on Applications of the Derivative to find the amounts spent on fertilizer and seed that will maximize profit. What is the maximum profit per acre? (Hint: Throughout this exercise you may ignore the coefficient of 0.01 until you need to find the maximum profit.)

(b)    Find the amounts spent on fertilizer and seed that will maximize profit using the first method shown in this chapter. (Hint: You will not need to use the budget constraint.)

(c)    Use the Lagrange multiplier method to solve the original problem.

(d)    Look for the relationships among these methods.