Maximum profit with MR equal to MC

Statement

Revenue is $R(q)=1300q-5q^{2}$ and cost is $C(q)=1500+80q$ (in €). Find the quantity that maximizes profit and the maximum profit.

Solution

Profit: $\pi(q)=R-C=1300q-5q^{2}-1500-80q=-5q^{2}+1220q-1500$. $\pi'(q)=-10q+1220=0\Rightarrow q=122$; since $\pi''=-10<0$ it is a maximum. $\pi(122)=-5\cdot 122^{2}+1220\cdot 122-1500=\boxed{72\,920\ \text{€}}.$