Question

Suppose that the price p​ (in dollars) and the weekly sales x​ (in thousands of​ units) of a certain commodity satisfy the demand equation 4p cubedplusx squaredequals38 comma 400. Determine the rate at which sales are changing at a time when xequals80​, pequals20​, and the price is falling at the rate of ​$.20 per week.

Answer 1

sales are increasing at the rate of 6000 units per week

Explanation:

Your demand equation appears to be ...

4p³ +x² =38400

Then differentiation gives ...

12p²·p' +2x·x' = 0

Solving for x', we get ...

x' = -6p²·p'/x

Filling in the given values, we find the rate of change of sales to be ...

x' = -6(20²)(-0.20/wk)/80 = 6/wk . . . . . in thousands of units/wk

Sales are increasing at the rate of 6000 units per week.