In a rainwater harvesting system, the rainfall from a flat roof that is 22 metres by 20 metres drains into a cylindrical tank with a base diameter of two metres and a height of three and a half metres. If the tank is full, find the rainfall in centimetres.
Let’s start with the diagram. We have a flat roof that is 22 metres by 20 metres. The rainwater from this roof drains off. The rainwater drains into this tank with a diameter of two metres and a height of three and a half metres. We’re looking for the rainfall in centimetres that would completely fill the cylindrical tank.
We know the volume of a cylinder equals 𝜋 times the radius squared times the height. And the volume of a rectangular prism equals the length times the width times the height. To solve this problem, we need to set the volume of the cylinder equal to the volume of the rectangular prism.
The volume of our tank, 𝜋𝑟 squared times its height, must be equal to the volume of the flat roof with rainwater on it. Plugging in what we know, we’ll keep 𝜋 as 𝜋. We’re given the diameter of our tank, and we know that the radius is half of the diameter. The radius of our tank is then one metre. The radius squared, one squared, times a height of three and a half is equal to 22 times 20. This ℎ represents the rainfall in centimetre. We want to know the depth of the rainfall. It’s our unknown.
On the left, three and a half times one squared equals three and a half — we’ll keep the 𝜋 term — equals 440 times ℎ. If we’re solving for ℎ, we need to get ℎ by itself. So we divide both sides by 440. The rainfall is equal to three and a half times 𝜋 times 440. At this point, we wanna make a substitution for 𝜋 to find an approximate value. We’ll use 22 over seven.
I noticed something else though about three and a half. I can write three and a half as a fraction, seven over two. We now have something like this: seven over two times 22 over seven all over 440. In the numerator, the seven over seven cancels out, and we can divide two by 22, leaving us with 11 over one. 11 over one equals 11. Our new fraction is 11 over 440. Both the numerator and the denominator here are divisible by 11. And that means our rainfall is one over 40.
But you have to remember that this is one 40th metre, because all of our dimensions were given in metres. Since our final answer is looking for rainfall in centimetres, we need to convert one 40th of a metre into centimetres. To do that, we multiply one 40th of a metre times 100 centimetres per metre. 100 divided by 40 simplifies to 10 over four, which simplifies to five over two. Five halves equals two and a half centimetres. To fill the tank, the rainfall in centimetres would have been two and a half centimetres.