- A company makes drinks containers out of metal.
The containers are modelled as closed cylinders with base radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\) and the capacity of each container is \(355 \mathrm {~cm} ^ { 3 }\)
The metal used
- for the circular base and the curved side costs 0.04 pence/ \(\mathrm { cm } ^ { 2 }\)
- for the circular top costs 0.09 pence/ \(\mathrm { cm } ^ { 2 }\)
Both metals used are of negligible thickness.
- Show that the total cost, \(C\) pence, of the metal for one container is given by
$$C = 0.13 \pi r ^ { 2 } + \frac { 28.4 } { r }$$
- Use calculus to find the value of \(r\) for which \(C\) is a minimum, giving your answer to 3 significant figures.
- Using \(\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} r ^ { 2 } }\) prove that the cost is minimised for the value of \(r\) found in part (b).
- Hence find the minimum value of \(C\), giving your answer to the nearest integer.