Answer
p=$175 - 0.01x - 0.01x^{2}$
Work Step by Step
R(x)= $\int (175 - 0.02x - 0.03x^{2})dx$
= $175x - 0.01x^{2} - 0.01x^{3} + C$
To find the C, we know R(0)=0 since if no items are sold, the revenue is 0. So:
$0 = 175(0) = 0.01(0)^{2} - 0.01(0)^{3} + C$
$C = 0$
Thus, the revenue function is: R(x) = $175x - 0.01x^{2} - 0.01x^{3}$
Recall that R= qp, where p is the demand function giving the price p as a function of q. Then
$175x - 0.01x^{2} - 0.01x^{3} = xp$
$\frac{175x - 0.01x^{2} - 0.01x^{3}}{x} = p$
The demand function is p=$175 - 0.01x - 0.01x^{2}$
