A manufacturer of car radios ships them to retailers in cartons of $n$ radios. The profit per radio is $\$59.50$, minus shipping cost of $\$25$ per carton, so the profit is $59.5n-25$ dollars per carton. To promote sales by assuring high quality, the manufacturer promises to pay the retailer $\$200X^2$ if $X$ radios in the carton are defective. Suppose radios are produced independently and that $5\%$ of radios are defective. How many radios should be packed per carton to maximize expected net profit per carton?
Suppose that $n$ radios are packed into a carton. Then, the expected profit is clearly $$\mu = 59.5n-25-200(0.05n)^2$$ Simplifying we get $$\mu = 59.5n-25-0.5n^2$$
We want to find a maximum, so we find a derivative:
Clearly there is just one maximum, so set $\mu'=0$ we find that $$n^2=59.5$$ and thus $$n\approx 7.7136\dots$$This however seems to be wrong. The textbook gives an answer of exactly $50$ in the solution without any explanation of the steps.