3 On a particular voyage, a ship sails 500 km at a constant speed of \(v \mathrm {~km} / \mathrm { h }\). The cost for the voyage is \(\pounds R\) per hour. The total cost of the voyage is \(\pounds T\).
- Show that \(T = \frac { 500 R } { v }\).
The running cost is modelled by the following formula.
$$R = 270 + \frac { v ^ { 3 } } { 200 }$$
The ship's owner wishes to sail at a speed that will minimise the total cost for the voyage. It is given that the graph of \(T\) against \(v\) has exactly one stationary point, which is a minimum.
- Find the speed that gives the minimum value of \(T\).
- Find the minimum value of the total cost.