2025 ACJC P1 Q6

Figure \(1\) shows an open cylindrical tank. Figure \(2\) shows the cross-sectional view of the tank. The external radius of the tank is \(r\text{ cm}\), the external height is \(h\text{ cm}\) and the tank is made with a material of thickness \(a\text{ cm}\) throughout. The internal volume of the tank is fixed at \(1000\pi\text{ cm}^3\).

1. Show that the volume of the material needed to make the cylindrical tank is given by \[V = k\pi \left[ \frac{r^2}{(r-a)^2} - 1 \right] + \pi r^2 a\text{, where}\hspace{0.5em} k \text{ is a constant to be determined.}\]

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2. Find, in terms of \(a\), the value of \(r\) that minimises \(V\). (You need not show that your answer gives a minimum.)[3]