2021 ACJC P1 Q10

The Gompertz differential equation provides a good model for gauging lung cancer growth. The differential equation is given as

\(\frac{\mathrm{d}V}{\mathrm{d}t}=aV\ln \left( \frac{K}{V} \right)\),

where \(V\)mm\(^{3}\) is the volume of the tumour at time \(t\) days after an early discovery, and \(a\) and \(K\) are positive real constants

1. Describe the behaviour of \(\frac{\mathrm{d}V}{\mathrm{d}t}\) as \(V\to K\).[1]
2. By using the substitution \(u=\ln \left( \frac{K}{V} \right)\), solve the given differential equation and show that \(V=K{{\mathrm{e}}^{-A{{\mathrm{e}}^{-at}}}}\), where \(A\) is an arbitrary constant.[5]
3. What happens to \(V\) as \(t\to \infty \)? State the significance of \(K\) in the context of this question.[2]

For the rest of this question, you may assume that \(a=0.01\) and \(K=8000\).

A patient was discovered early to have a lung tumour of volume \(100\)mm\(^{3}\).

4. Find the size of the tumour after \(100\)days. Give your answer to the nearest mm\(^{3}\).[2]
5. Sketch the graph of \(V\) against \(t\).[2]