2026 EJC P1 Q7

It is given that a sphere of radius \(r\) has surface area \(4\pi r^2\) and volume \(\frac43\pi r^3\).

An industrial engineer is designing a storage tank in the shape of a closed composite solid which consists of three parts:

• the curved surface of a hemisphere of radius \(r\) m;
• the curved surface of a cylinder of radius \(r\) m and height \(h\) m;
• the circular top of radius \(r\) m.

The total volume of the tank is fixed at \(k\text{ m}^3\), where \(k\) is a positive constant. You can assume that the wall of the tank is of negligible thickness.

During the preliminary design phase, the engineer uses a simulation software to study the tank’s proportions. The software models \(r\) increasing at a constant rate of \(1\text{ m s}^{-1}\), while the volume remains at \(k\text{ m}^3\) and the composite structure of the tank is kept intact.

1. Find \(\frac{\mathrm{d}h}{\mathrm{d}t}\) at the instant when \(r=\sqrt[3]{\frac{6k}{7\pi}}\).[4]

The total external surface area is given as \(S\text{ m}^2\).

2. Show that \(S=\frac53\pi r^2+\frac{2k}{r}\).[3]
3. Find the value of \(r\), in terms of \(k\), for which \(S\) is a minimum value.[4]