Answer
$C=-200x+620.$
$P=-400x^{2}+1400x-620$.
At ${{\$}} 1.75\quad $ per log-on, the maximum profit is ${{\$}} 605$ per month.
Work Step by Step
As a function of $q$, monthly cost is
$C=0.5q+20$.
To express C as a function of x, substitute $q=-400x+1200$:
$C=0.5(-400x+1200)+20$
$C=-200x+620$.
The profit is$ \quad P=R-C$
$P=xq-C$
$P=-400x^{2}+1400x-620$.
The graph of P is a parabola that opens down.
The profit is largest at the vertex, when $x=-b/(2a)$
$x=-\displaystyle \frac{1400}{-2\times 400}= {{\$}} 1.75\quad $ per log-on.
The corresponding profit is
$P=-400(1.75)^{2}+1400(1.75)-620$
$P={{\$}} 605$ per month.