Medical equipment is sterilised by placing it in a hot oven for a time and then removing it and letting it cool for the same time. The equipment at temperature warms and cools at a rate equal to the product of a constant and the difference between its temperature and its surroundings, when warming in the oven and when cooling outside. The equipment starts the sterilisation process at temperature .
Bacteria are killed by the heat treatment. Their number decreases at a rate equal to the product of the current number and a destruction factor . This destruction factor varies linearly with temperature, vanishing at and having a maximum at .
Find an implicit equation for such that the number of bacteria is reduced by a factor of by the sterilisation process.
A second hardier species of bacteria requires the oven temperature to be increased to achieve the same destruction factor . How is the sterilisation time affected?