= Solution
Assume $C,Q>0$. The expected net profit per item is $\Pi(\alpha)=Q\mathbb E[T]-C/\alpha^2=Q/\alpha-C/\alpha^2$. Put $u=1/\alpha>0$ and complete the square:
$$
\Pi=Qu-Cu^2=\frac{Q^2}{4C}-C\left(u-\frac Q{2C}\right)^2.
$$
This <quadratic function> has a unique global maximum at $u=Q/(2C)$. Hence
$$
\boxed{\alpha_{\rm opt}=\frac{2C}{Q},\qquad\Pi_{\max}=\frac{Q^2}{4C}.}
$$
The calculation uses the expected operating time, rather than replacing lifetime by an arbitrary fixed time.
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