For a price and cost function, at what rate do the weekly sales change per week?

Question

For a price function of $p=4000-25x$ and a cost function of $C=1800x+4500$, if the profit is increasing at a rate of $3000$ per week and the weekly sales are $x=32$ units, at what rate do the weekly sales change per week?

Answer 1

Hint:

Profit = Revenue - Cost

$\frac{dP}{dT}$ = $\frac{dP}{dx}$*$\frac{dx}{dT}$ ---- (1)

P = p(x)*x - c(x) = (4000-25x)*x - (1800x+4500) = 4000x - 25$x^2$ - 1800x - 4500

$\frac{dP}{dT}$ = (2200 - 50x)*$\frac{dx}{dT}$

You know the left hand side of equation (1) to be 3000

3000 = (2200 - 50*32)*$\frac{dx}{dT}$

Find $\frac{dx}{dT}$ (= 5) - If you consider x to be the sales, then the answer is 5, if you consider sales (to be in dollars then the below steps need to be added)

Now, $\frac{dS}{dT}$ = $\frac{dS}{dx}$*$\frac{dx}{dT}$

$\frac{dS}{dT}$ = (4000 - 50x)*5 = (4000 - 50*32)*5 = $12,000

Sorry Kaine: I waited for a long time before the OP could respond but I am helping her understand the concept so that she becomes comfortable after this.