<p>A researcher is investigating the spread of a certain disease in a town with a population of $3000$ people. The researcher suggests that $I$, the number of people infected by the disease at time $t$ days satisfies the differential equation $\frac{\mathrm{d}I}{\mathrm{d}t}=kI\left( 3000-I \right)$, where $k$ is a positive constant.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Given that $I=30$ when $t=0$, show that $I=\frac{3000}{1+99{{e}^{-3000kt}}}$.</div><p style="text-align: right;">[6]</p></li><li><div class="list-item-paragraph">It is further observed that $I=240$ when $t=7$, find the time it takes for $90\%$ of the population to be infected by the disease.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">State, in the context of this question, one assumption needed to model the spread of the disease in the town by the given differential equation.</div><p style="text-align: right;">[1]</p></li></ol>

A researcher is investigating the spread of a certain disease in a town with a population of $3000$ people. The researcher suggests that $I$, the number of people infected by the disease at time $t$ days satisfies the differential equation $\frac{\mathrm{d}I}{\mathrm{d}t}=kI\left( 3000-I \right)$, where $k$ is a positive constant.

Given that $I=30$ when $t=0$, show that $I=\frac{3000}{1+99{{e}^{-3000kt}}}$.

It is further observed that $I=240$ when $t=7$, find the time it takes for $90\%$ of the population to be infected by the disease.

State, in the context of this question, one assumption needed to model the spread of the disease in the town by the given differential equation.

2013 JJC P2 Q2

2013 JJC P2 Q2

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