9. Bacteria are growing on the surface of a dish in a laboratory.
The area of the dish, \(A \mathrm {~cm} ^ { 2 }\), covered by the bacteria, \(t\) days after the bacteria start to grow, is modelled by the differential equation
$$\frac { \mathrm { d } A } { \mathrm {~d} t } = \frac { A ^ { \frac { 3 } { 2 } } } { 5 t ^ { 2 } } \quad t > 0$$
Given that \(A = 2.25\) when \(t = 3\)
- show that
$$A = \left( \frac { p t } { q t + r } \right) ^ { 2 }$$
where \(p , q\) and \(r\) are integers to be found.
According to the model, there is a limit to the area that will be covered by the bacteria.
- Find the value of this limit.
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