A pencil holder is in the shape of an open circular cylinder of radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The surface area of the cylinder (including the base) is \(250 \mathrm {~cm} ^ { 2 }\).
Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cylinder is given by \(V = 125 r - \frac { \pi r ^ { 3 } } { 2 }\).
Use calculus to find the value of \(r\) for which \(V\) has a stationary value.
Prove that the value of \(r\) you found in part (b) gives a maximum value for \(V\).
Calculate, to the nearest \(\mathrm { cm } ^ { 3 }\), the maximum volume of the pencil holder.