Monopolist's optimal choice

Statement

A monopolist has demand $p=200-4q$ (in €) and total cost $C(q)=500+20q$. Find the quantity and price that maximize profit, and the maximum profit.

Solution

Revenue: $R=pq=200q-4q^{2}$. Profit: $\pi=R-C=-4q^{2}+180q-500$. $\pi'(q)=-8q+180=0\Rightarrow q=22{.}5$ (maximum, $\pi''=-8<0$). Price: $p=200-4\cdot 22{.}5=110\ \text{€}$. $\pi=-4\cdot 22{.}5^{2}+180\cdot 22{.}5-500=\boxed{1525\ \text{€}}.$