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 \(r\) cm, the external height is \(h\) cm and the tank is made with a material of thickness \(a\) cm throughout. The internal volume of the tank is fixed at \(1000\pi\) cm\(^3\).

1. Show that the volume of the material needed to make the cylindrical tank is given by \[V = k\pi \left[ \frac{r^2}{(r - a)^2} - 1 \right] + \pi r^2 a\] where \(k\) is a constant to be determined.[3]
2. Find, in terms of \(a\), the value of \(r\) that minimises \(V\). (You need not show that your answer gives a minimum.)[3]