 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2005/paper-1/4/b/ii/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

Assume

C, Q > 0

. The expected net profit per item is

Π (α) = Q E [T] - C / α^{2} = Q / α - C / α^{2}

. Put

u = 1/ α > 0

and complete the square:

Π = Q u - C u^{2} = \frac{Q ^{2}}{4 C} - C (u - \frac{Q}{2 C})^{2} .

u = Q / (2 C)

. Hence

α_{opt} = \frac{2 C}{Q}, Π_{m a x} = \frac{Q ^{2}}{4 C} .

The calculation uses the expected operating time, rather than replacing lifetime by an arbitrary fixed time.

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