6. A container made from thin metal is in the shape of a right circular cylinder with height \(h \mathrm {~cm}\) and base radius \(r \mathrm {~cm}\). The container has no lid. When full of water, the container holds \(500 \mathrm {~cm} ^ { 3 }\) of water.

1. Show that the exterior surface area, \(A \mathrm {~cm} ^ { 2 }\), of the container is given by \(A = \pi r ^ { 2 } + \frac { 1000 } { r }\).
2. Find the value of \(r\) for which \(A\) is a minimum.
3. Prove that this value of \(r\) gives a minimum value of \(A\).
4. Calculate the minimum value of \(A\), giving your answer to the nearest integer.