Solve the differential equation
$$\frac { \mathrm { d } x } { \mathrm {~d} t } = - \frac { 1 } { 5 } ( x + 1 ) ^ { \frac { 1 } { 2 } }$$
given that \(x = 80\) when \(t = 0\). Give your answer in the form \(x = \mathrm { f } ( t )\).
A fungus is spreading on the surface of a wall. The proportion of the wall that is unaffected after time \(t\) hours is \(x \%\). The rate of change of \(x\) is modelled by the differential equation
$$\frac { \mathrm { d } x } { \mathrm {~d} t } = - \frac { 1 } { 5 } ( x + 1 ) ^ { \frac { 1 } { 2 } }$$
At \(t = 0\), the proportion of the wall that is unaffected is \(80 \%\). Find the proportion of the wall that will still be unaffected after 60 hours.
A biologist proposes an alternative model for the rate at which the fungus is spreading on the wall. The total surface area of the wall is \(9 \mathrm {~m} ^ { 2 }\). The surface area that is affected at time \(t\) hours is \(A \mathrm {~m} ^ { 2 }\). The biologist proposes that the rate of change of \(A\) is proportional to the product of the surface area that is affected and the surface area that is unaffected.
Write down a differential equation for this model.
(You are not required to solve your differential equation.)
A solution of the differential equation for this model is given by
$$A = \frac { 9 } { 1 + 4 \mathrm { e } ^ { - 0.09 t } }$$
Find the time taken for \(50 \%\) of the area of the wall to be affected. Give your answer in hours to three significant figures.
(4 marks)