2025 YIJC P1 Q3

A closed cylindrical tank with radius \(r\) m and height \(h\) m is to be constructed to hold \(1000\) m\(^3\) of water. The material for the base and lid costs \(50\) per m\(^2\), while the material for the curved surface costs \(30\) per m\(^2\).

1. Show that the total cost \(C\) (in dollars) can be expressed as \(C = 100\pi r^2 + \frac{60000}{r}\).[2]
2. Use differentiation to find the exact radius of the cylinder for which \(C\) is minimum. State the minimum cost.[4]
3. Sketch the graph showing the total cost as the radius of the tank varies.[2]

A change to one of the materials used results in the cylinder's volume doubling to \(2000\) m\(^3\), while maintaining the same minimum cost and radius found in part (b).

4. State which material's cost has changed and its value.[1]