# Paper 2, Section II, A

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

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

Find an implicit equation for $T$ such that the number of bacteria is reduced by a factor of $10^{-20}$ by the sterilisation process.

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