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CMA Final · Strategic Cost Management · Business Application of Maxima and Minima

A company must make an open-top rectangular tank with a square base and a volume of 32 cubic metres. The base and sides are made from the same material. What base side length minimises the material (surface area) used?

With height 32 divided by a squared, surface area is a squared plus 128 over a. Setting the derivative to zero gives a cubed equal to 64, so the base side is 4 metres, with height 2 metres and minimum area 48 square metres.

  1. A2 metres
  2. B4 metresCorrect
  3. C8 metres
  4. D3 metres

Explanation

Let base side be a and height h = 32/a^2. Area S = a^2 + 4ah = a^2 + 128/a. dS/da = 2a - 128/a^2 = 0 gives a^3 = 64, so a = 4. Then h = 2 and S = 16 + 32 = 48 sq m. Second derivative 2 + 256/a^3 is positive, so it is a minimum. A 2-metre base mistakenly equals the height.

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