Show formula then optimise: cost/non-geometric objective

Questions where the quantity to be optimised is a cost, revenue, or other non-purely-geometric quantity (e.g. polishing cost per cm²), requiring showing a cost/objective formula then optimising it.

1 questions · Standard +0.3

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Edexcel C2 2015 June Q9
10 marks Standard +0.3
9. A solid glass cylinder, which is used in an expensive laser amplifier, has a volume of \(75 \pi \mathrm {~cm} ^ { 3 }\).
The cost of polishing the surface area of this glass cylinder is \(\pounds 2\) per \(\mathrm { cm } ^ { 2 }\) for the curved surface area and \(\pounds 3\) per \(\mathrm { cm } ^ { 2 }\) for the circular top and base areas. Given that the radius of the cylinder is \(r \mathrm {~cm}\),
  1. show that the cost of the polishing, \(\pounds C\), is given by $$C = 6 \pi r ^ { 2 } + \frac { 300 \pi } { r }$$
  2. Use calculus to find the minimum cost of the polishing, giving your answer to the nearest pound.
  3. Justify that the answer that you have obtained in part (b) is a minimum.