IB Calculus Problem 002
A closed cylindrical metal box has a radius of \(r\) centimetres and a height of \(h\) centimetres, with a volume of \(20\pi\, cm^3\).
The figure is not to scale.
- Express \(h\) as a function of \(r\).
The metal for the base and lid of the box costs 10 cents per \(cm^2\) and the metal for the curved side costs \(8\) cents per \(cm^2\).
The total cost of the metal, in cents, is \(C\).
- Show that \(C\,=\,20\pi{}r^2 + \frac{320\pi}{r}\)
- Given that a minimum value of \(C\) exists, find that minimum value in terms of \(\pi\).