WORLD OF FIGURES: How to construct a water tank with minimum materials

In August 2017, a reader asked me to help him work out how much fencing wire he would need to fence a quarter acre parcel of land. I explained that the answer depends more on the shape of the plot than on its area.

This problem was somewhat similar to one that I got from John Mungai recently. He called me asking for help to work out how much construction material would be required to build a 100,000-litre water tank.

Just like in the case of fencing a parcel of land, the answer here also depends more on the shape than on the volume of the container. The important parameter that determines the quantity of material is the surface area of the tank. This is analogous to the perimeter in the case of a plot.

Now, for a given area of land, the shape of land that has the shortest perimeter is a circle. But a circular plot is impractical because circles do not tessellate. In the same breath, for a fixed volume of container, a sphere would have the smallest surface area.

But a sphere is unstable; it does not stand stationary with ease. In addition, it can easily collapse when filled with water — 100,000 litres weigh 100 tonnes!

The best shape for a water tank is a cylinder. It is efficient in material consumption and its flat base is very stable. In addition, the curved circular sides distribute the pressure from the inside evenly so it does not collapse as easily as a sphere or even a cuboid would.

The capacity of a cylindrical tank is simply the product of its base area and its height. There is a small challenge, however, in finding out which height would result from the smallest surface area for a given volume.

It turns out that the best ratio is 1-to-1; that is, the tank will have least surface area if its height is equal to its diameter.

So now, John’s question is easy: all we need to do is find out what height to give a tank of 100,000 litres.

The answer comes out as slightly more than 5m. If you can remember your primary school math, you should be able to work out the volume of a cylindrical tank whose base diameter is 5m and height is also 5m. It is 98,125 litres.

John wants to construct his tank using plastic sheets reinforced with steel meshing. So, he needs to know the size of plastic material required. This works out to 118 square metres.

But since the base of the tank is circular, there will be some waste off-cuts; perhaps 10 per cent of the ideal area.

Therefore, I think, about 130 square metres of plastic sheeting should be enough.

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