Paper 2, Section II, A

Differential Equations | Part IA, 2013

Medical equipment is sterilised by placing it in a hot oven for a time TT and then removing it and letting it cool for the same time. The equipment at temperature θ(t)\theta(t) warms and cools at a rate equal to the product of a constant α\alpha and the difference between its temperature and its surroundings, θ1\theta_{1} when warming in the oven and θ0\theta_{0} when cooling outside. The equipment starts the sterilisation process at temperature θ0\theta_{0}.

Bacteria are killed by the heat treatment. Their number N(t)N(t) decreases at a rate equal to the product of the current number and a destruction factor β\beta. This destruction factor varies linearly with temperature, vanishing at θ0\theta_{0} and having a maximum βmax\beta_{\max } at θ1\theta_{1}.

Find an implicit equation for TT such that the number of bacteria is reduced by a factor of 102010^{-20} by the sterilisation process.

A second hardier species of bacteria requires the oven temperature to be increased to achieve the same destruction factor βmax\beta_{\max }. How is the sterilisation time TT affected?

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