a) Use the substitution \(u = 4 – \sqrt{s}\) to show that

\[\int_{}^{}\frac{\text{dh}}{4 – \sqrt{s}} = – 8\ln\left| 4 – \sqrt{s} \right| – 2\sqrt{s} + k\]

where *k* is a constant

[6 marks]

A coral reef is growing back after global temperatures are reduced from their peak value.

The rate of change of area covered by the reef is modelled by the differential equation

\[\frac{\text{ds}}{\text{dt}} = \frac{t^{0.25}(4 – \sqrt{s})}{20}\]

Where *s* is the surface area of the reef in m^{2} and *t* is the time, in years, after the reef begins to regrow.

b) Find, according to the model, the range of areas that could be covered by the coral reef.

[2 marks]

The coral reef has a surface area of 1m^{2} when it starts to regrow.

According to the model,

c) Calculate the time this reef would take to cover 12 m^{2}, giving your answer to 3 significant figures.

[7 marks]