I wanted to find out how tall the rugby posts at my local stadium were. Taking a handy rod as my standard unit of length, I went out to the field with the rod and a gadget for measuring angles from ground level.

I walked out 10 rods from one of the goal posts and measured the angle from the ground to the top of the post. Then, just to be certain my calculations would be as accurate as possible, I walked another 10 rods in the same direction, and measured the angle again. To be absolutely precise (as I'm a bit of a perfectionist), I walked a final 10 rods in the same direction and measured the angle a third time.

When I went home to calculate the height of the goal post, I was surprised to discover that the sum of my three angles was precisely a right angle.

How tall were the goal posts in rods?

let the angles be a,b and c respectively and the goal post x.

then tan a = x/10, tan b = x/20 and tan c = x/30;

Using  trig, we have

tan(a + b) = 30x/(200 - x^2) 

Adding tan c gives

tan(a + b + c)(1 - (x^2/(200 - x^2)))

tan(a + b + c) = tan 90 = sin 90/cos 90

finally we have

(1100x - x^3)/30 = (sin 90/cos 90)(200 - 2x^2)

which gives x = 10

a = 45, b = 26.565, c = 18.435

Posted by Nebo on 2005-03-13 18:56:05