William W. answered • 12/09/19

Math and science made easy - learn from a retired engineer

Profit = Revenue - Cost so

P(x) = (32x - 0.8x^{2}) - (5x + 12)

P(x) = -0.8x^{2} + 27x - 12

To find the rate of change per time (per day actually) we can take the derivative with respect to time:

dP/dt = -1.6x•(dx/dt) + 27•(dx/dt)

We are being asked to find "the rate of change of profit with respect to time" which is dP/dt

We are give x = 15 and "the rate of change of production" or dx/dt = 8

So dP/dt = -1.6(15)(8) + 27(8) = 24

The units of profit is $

The units of time is days

So the answer is $24/day