Bacteria are growing on the surface of a dish in a laboratory. The area of the dish, $$\(A \mathrm{~cm}^{2}\)$$, covered by the bacteria, $$\(t\)$$ days after the bacteria start to grow, is modelled by the differential equation $$\[ \frac{\mathrm{d} A}{\mathrm{~d} t}=\frac{A^{\frac{3}{2}}}{5 t^{2}} \quad t> 0 \]$$ Given that $$\(A=2.25\)$$ when $$\(t=3\)$$ (a) show that $$\[ A=\left(\frac{p t}{q t+r}\right)^{2} \]$$ where $$\(p, q\)$$ and $$\(r\)$$ are integers to be found. (7) 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. (2)

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