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 angle formed at a distance of 30 rods from the goal post be 'a', 20
rods be 'b' and 10 rods be 'c'. Let the height of the goal post be H
rods. Let 'r' represent one rod length as a unit.
Therefore, height of goal post = H r.
And we get,
tan a = (H r) / (30 r)
tan b = (H r) / (20 r)
tan c = (H r) / (10 r)
This in turn gives:
H = 30 tan a = 20 tan b = 10 tan c
We get the equality:
3 tan a = 2 tan b = tan c
Also we are given that
a + b + c = 90 (in degrees)
Therefore, we get:
tan a + tan b + tan c - (tan a . tan b . tan c)
---------------------------------------------------- = infinity
1 - tan a . tan b - tan b . tan c - tan c. tan a
From the above we get:
1 - tan a . tan b - tan b . tan c - tan c . tan a = 0
tan a = tan c / 3
tan b = tan c / 2
1 - (tan c) ^ 2 = 0
Therefore, tan c = ± 1
Taking tan c = +1, we get c = 45 degrees
Substituting this value of c we get,
a = 18.435 deg
b = 26.565 deg
Therefore, the three angles are : (18.435, 26.565, 45.0) degrees
Posted by Ujjawal
on 2005-03-11 11:10:34