Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 331 / 1 / f

Table of contents
- Solution f

Solution

Put

x = k^{2}

and

T = λ / σ

. Differentiating

R a

gives the exact stationarity equation

(π^{2} + x)^{2} (2 x - π^{2}) = π^{2} T^{2} .

When

T ≫ 1

, the optimum has

x ≫ π^{2}

, so

2 x^{3} \sim π^{2} T^{2}

. Therefore

k \sim (\frac{π ^{2}}{2})^{1/6} (\frac{λ}{σ})^{1/3} .

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