Past exam of the mathematics course of the University of Cambridge / 2016 / ia / Paper 2 / 5A / b / iv / Solution

Impose the target temperature on the post-jump exponential decay:

[(\frac{α}{2} - 1) T_{0} + β] e^{- k (t_{2} - t_{1})} = (α - 1) T_{0} .

Solving for the jump and eliminating

t_{1}

using

e^{k t_{1}} = 2 (α - 1) / (α - 2)

gives

β = (α - 1) T_{0} e^{k (t_{2} - t_{1})} - \frac{α - 2}{2} T_{0} = \frac{α - 2}{2} T_{0} (e^{k t_{2}} - 1) .

This is the unique required temperature increase, and it satisfies the stipulated lower bound. Indeed

t_{2} > t_{1}

implies

β - \frac{α T _{0}}{2} = (α - 1) T_{0} (e^{k (t_{2} - t_{1})} - 1) > 0.

Thus no additional restriction on the prescribed time is needed.

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