The strapline says it all. I'm including this page simply as a counterblast to all of those (including the prospective user of this ramp) who say "What's the point of maths?". The maths itself is unspectacular - it's the everyday usefulness of it that I want to emphasise.
The problem
My son Matt wanted a short ramp for practising skateboard and rollerblade tricks on. A simple inclined plane is no good - the transition from the flat ground to the sloped surface is too jerky. The board therefore needs a surface which is roughly tangential to the ground where it touches the ground; making a curved surface is easy using thin plywood (well supported). However, sometimes you want to approach the ramp "the wrong way" and jump onto the high end, and the impact is more than thin plywood can take. Therefore the ramp had a curved transition section at the bottom, and a simple inclined plane at the top (made from thick plywood). You can see an exaggerated profile of the ramp in the diagram below.
At first I thought that the geometrical design of the board would be simple. But after playing around a bit I realised that it was quite complicated. The curved section has to be tangential not only to the ground (at its lower end) but also to the flat part of the ramp (at its upper end). At the same time the top of the board has to have a predetermined height, and of course we want to fix the length of the board. Working it all out by eye was not going to work.
The solution
Therefore I did the maths below, and derived an expression for the height of the top of the board as a function of the length of the board and the angle θ subtended by the curved section at its centre of curvature. Inserting a value for the length of the board, we can then vary the angle θ until we get the height we want. This is easy with a spreadsheet. To simplify things I decided that the curved and straight sections would each occupy (horizontally) half the length of the whole ramp.
It worked. Therefore maths is not useless. QED.
