Optimise cost or profit model

5. A company makes a particular type of watch. The annual profit made by the company from sales of these watches is modelled by the equation $$P = 12 x - x ^ { \frac { 3 } { 2 } } - 120$$ where \(P\) is the annual profit measured in thousands of pounds and \(\pounds x\) is the selling price of the watch. According to this model,

1. find, using calculus, the maximum possible annual profit.
2. Justify, also using calculus, that the profit you have found is a maximum.

1. A diesel lorry is driven from Birmingham to Bury at a steady speed of v kilometres per hour. The total cost of the journey, \(\pounds C\), is given by

$$C = \frac { 1400 } { v } + \frac { 2 v } { 7 } .$$

1. Find the value of \(v\) for which \(C\) is a minimum.
2. Find \(\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} v ^ { 2 } }\) and hence verify that \(C\) is a minimum for this value of \(v\).
3. Calculate the minimum total cost of the journey.

1. A lorry is driven between London and Newcastle.

In a simple model, the cost of the journey \(\pounds C\) when the lorry is driven at a steady speed of \(v\) kilometres per hour is $$C = \frac { 1500 } { v } + \frac { 2 v } { 11 } + 60$$

1. Find, according to this model,
   1. the value of \(v\) that minimises the cost of the journey,
   2. the minimum cost of the journey.
      (Solutions based entirely on graphical or numerical methods are not acceptable.)
2. Prove by using \(\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} v ^ { 2 } }\) that the cost is minimised at the speed found in (a)(i).
3. State one limitation of this model.

1. 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.

1. Show that the total cost, \(C\) pence, of the metal for one container is given by $$C = 0.13 \pi r ^ { 2 } + \frac { 28.4 } { r }$$
2. Use calculus to find the value of \(r\) for which \(C\) is a minimum, giving your answer to 3 significant figures.
3. 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).
4. Hence find the minimum value of \(C\), giving your answer to the nearest integer.

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 \(\pounds T\).

1. 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.
2. Find the speed that gives the minimum value of \(T\).
3. Find the minimum value of the total cost.