10. A solid right circular cylinder has radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\).
The total surface area of the cylinder is \(800 \mathrm {~cm} ^ { 2 }\).
- Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cylinder is given by
$$V = 400 r - \pi r ^ { 3 }$$
Given that \(r\) varies,
- use calculus to find the maximum value of \(V\), to the nearest \(\mathrm { cm } ^ { 3 }\).
- Justify that the value of \(V\) you have found is a maximum.
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