2025 SCGS PRELIMS P1 Q10

The diagram shows a container which consists of an open cylinder of radius \(r\) m and height \(h\) m joined to a hemispherical bottom of radius \(r\) m. The container is to be made of thin metal sheets of negligible thickness. The volume of the cylindrical part of the container is \(300\ \text{m}^3\). [Surface area of a sphere of radius \(r\) is \(4\pi r^2\).]

1. Express \(h\) in terms of \(r\).[1]
2. The manufacturer bought metal sheets at \(\$60\) per square metre. Show that the total cost, \(C\) dollars, of the metal sheets required to make the container is given by \[ C = 120\pi r^2 + \frac{36000}{r}. \][2]
3. Given that \(r\) can vary, find the stationary value of \(C\) and determine its nature.[6]