N2021 P1 Q10

Scientists model the number of bacteria, \(N\), present at a time \(t\) minutes after setting up an experiment. The model assumes that, at any time \(t\), the growth rate in the number of bacteria is \(kN\), for some positive constant \(k\). Initially there are \(100\) bacteria and it is found that there are \(300\) at time \(t=2\).

1. Write down and solve a differential equation involving \(N\), \(t\) and \(k\). Find \(k\) and the time it takes for the number of bacteria to reach \(1000\).[4]

The scientists repeat the experiment, again with an initial number of \(100\) bacteria. The growth rate, \(kN\), for the number of bacteria is the same as that found in part (a). This time they add an anti-bacterial solution which they model as reducing the number of bacteria by \(d\) bacteria per minute.

2. Write down and solve a differential equation, giving \(t\) in terms of \(N\) and \(d\). Hence find \(N\) in terms of \(t\) and \(d\).[5]
3. 1. Find the range of values of \(d\) for which the number of bacteria will decrease.[1]
   2. In the case where \(d=58\), find the time taken for the number of bacteria to reach zero.[2]