The cube of a number, x^{3}, is what you get when
you multiply a number by itself and then by itself again. For example 4^{3}
equals 4 * 4 * 4 which equals 64. Rather like its sibling
x^{2},
it's easy to see where x^{2} gets its name from.
If you want to work out the volume of a cube that's 3 metres
wide then 3 cubed ( = 3 * 3 * 3 = 27) is the number of cubic
metres you're looking for. So the word "cube" crops up straight away.
I'm not denying that x^{3} is very useful for working out volumes like this.
It doesn't need to be a cubist kind of shape either. You might know the famous formula for
the volume of a sphere: 4/3 pi * r^{3} (where
r is the radius of the sphere). No matter what 3D shape you choose, if you scale it up evenly
by a factor of 2 then it's volume will increase by a factor of 8 (because 2^{3} = 8).
But there are hundreds of other uses for x^{3} that have nothing to do
with shapes and volumes.
For instance, here's our bass player, Doug, with his motorcycle. Now not a lot people know
this, but the number of horsepower (or kilowatts, for you metric types) that you need
to overcome the wind resistance against a bike goes up with the cube of
your speed. So you would need 8 times as much power to go twice as fast.
To triple your speed you would need a staggering 27 times as much power.
The wind pressure acting on you and the bike only goes up by x squared.
when you multiply the speed by a cetain factor x. So doubling your speed makes gives
you a four fold increase in wind drag. Thus is discussed on the x^{2} page.
There, we were talking about wind rushing past a stationary object (the nightclub sign)
rather than a moving object (the bike) rushing through stationary air. But the principle
is the same. But if the bike starts going twice as fast then it also goes twice as
far every second. So the engine has to push the bike four times as hard
and two times as far. This takes 4 * 2 = 8 times as much energy every
second, which also means that eight times as much petrol has to be burned every second.
So in the general case, where you want to multiply your speed by some factor of your choice,
x, then you have to push the bike x^{2} time harder and x
times further so you'll need an engine that is
x^{2} * x = x^{3} times
as powerful!
