Problem

Bacteria are growing on the surface of a dish in a laboratory.

The area of the dish, $A$ cm$^2$, covered by the bacteria, $t$ days after the bacteria start to grow, is modelled by the differential equation

$$\frac{dA}{dt}=\frac{A^{3/2}}{5t^2}\qquad t>0$$

Given that $A=2.25$ when $t=3$

(a) show that

$$A=\left(\frac{pt}{qt+r}\right)^2$$

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.

(b) Find the value of this limit.