Answer
p=$500- 0.1x^{\frac{1}{2}}$
Work Step by Step
R(x) = $\int(500 - 0.15\sqrt x)dx = \int(500 - 0.15x^{\frac{1}{2}})dx$
=$ 500x -0.1 x^{\frac{3}{2}} + C$
We know R(0)=0 since if no items are sold, the revenue is 0. So:
$0= 500(0) -0.1 (0)^{\frac{3}{2}} + C$
$C=0$
Thus, the revenue function is: R(x)= $ 500x -0.1 x^{\frac{3}{2}}$
Recall that R= xp, where p is the demand function giving the price p as a function of x. Then
$ 500x -0.1 x^{\frac{3}{2}}$=xp
$\frac{ $ 500x -0.1 x^{\frac{3}{2}}}{x}$=p
The demand function is p=$500- 0.1x^{\frac{1}{2}}$
