- A company decides to manufacture a soft drinks can with a capacity of 500 ml .
The company models the can in the shape of a right circular cylinder with radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\).
In the model they assume that the can is made from a metal of negligible thickness.
- Prove that the total surface area, \(S \mathrm {~cm} ^ { 2 }\), of the can is given by
$$S = 2 \pi r ^ { 2 } + \frac { 1000 } { r }$$
Given that \(r\) can vary,
- find the dimensions of a can that has minimum surface area.
- With reference to the shape of the can, suggest a reason why the company may choose not to manufacture a can with minimum surface area.