A manufacturer has determined that the weekly profit from the sale of x items is given by the function below. It is estimated that after t days in an week, x items will have been produced. Find the rate of change of profit with respect to time at the end of 7 days.

Question

P9x) = -x^2+600x-3000 with x=1.5t^2-2t

First I tried plugging in "t" to the x equation, then "x" into the p equation, which didn't produce the right answer. Then I tried finding the derivative of the p equation and plugging in t to x and x into the derivative, which also didn't produce the correct answer

Answer 1

P(x) = -x^2+600x-3000
x(t) = 1.5t^2-2t

dP/dt = dP/dx * dx/dt
= (-2x+600)(3t-2)
= (-2(1.5t^2-2t))(3t-2)
= -9t^3+18t^2-8t
= -2261

or,

dP/dt = (-2x+600)(3t-2)
= (-119)(19)
= -2261