Watching the Rugby World Cup we were struck by the number of penalties that are awarded on the 15 metre line. [There are reasons for this, but they aren't relevant to this puzzle.]
An oddity of Rugby is that you don't have to take the penalty from the place it is awarded; you can go back towards your own try line, parallel to the touchline. Players do this quite often to "give themselves a better angle" for a kick at goal.
This set us thinking - what is the optimum position along the 15 metre line from which to take a kick at goal. This ought to be the sort of information every Rugby goal kicker should know.
A Rugby pitch is 70 metres wide. The 15 metre lines are 15 metres from the sides of the pitch. And the posts are 5.6 metres apart. Here is a pitch diagram.
So where is the point on the 15 metre line that is best for kicking at goal?
A bit of arithmetic will show that the distance between a 15 metre line and the near post is 17.2 metres and the distance between a 15 metre line and the far post is 22.8 metres.
As the ball is moved along the 15 metre line there are two right-angled triangles formed. Both have sides along the goal line and the 15 metre line, with their respective hypotenuses the lines to the near and far posts respectively.
If the angle to the near post is a, the angle to the far post b, and the distance between the ball and the goal line x.
Tan (a) = 17.2 / x and Tan (b) = 22.8 / x
a = Arctan (17.2 / x) and b = Arctan (22.8 / x)
And the angle between the posts is b - a.
Feeding in various values for x the largest value for b - a comes when x is about 19.8 metres. So just inside the 22 metre line is about right.