An open-topped fish tank is to be made for an aquarium. It will have a square base, rectangular sides, and a volume of 60 m3. The base materials cost £15 per m2 and the sides £8 per m2. What should the height be to minimise costs?

Establish the areas of each of the five faces of the aquarium in terms of x and y, and then sum to obtain a function of the cost in x and y. Then identify that the volume also gives a function in x and y. Rearrange this function into y = f(x), and substitute this into our area function. Differentiate this and set equal to 0 to find x=4 and y=3.75. Differentiate again in order to show that x=4 does result in a minimum.