During a gale a maypole was broken in such a manner that it struck the ground at a distance twenty feet from the base of the pole.

It was repaired and later broke a second time at a point five feet lower. This time it struck the ground thirty-five feet from the base.

What was the original height of the pole?

The original height of the pole is about 82.5 feet.

Each time the pole breaks, it forms a right triangle with the ground. For the first triangle, call the side segment x and the hypotenuse segment y. For the second triangle, the side is x-5 and the hypotenuse is y+5.

Applying Pythagorean theorem (a^2 + b^2 = c^2, where c is the hypotenuse) to the two triangles yields the following equations:

400 + x^2 = y^2
1225 + x^2 - 10x +25 = y^2 +10y +25

Substituting the y's in the second equation and simplifying leaves:

1225 + x^2 - 10x +25 = 400 + x^2 + 10*SQRT(400 + x^2) + 25
825 - 10x = 10*SQRT(400 + x^2)

Square the equation:

680625 - 16500x + 100x^2 = 40000 + 100x^2
16500x = 640625
x = 38.82575757...

Putting x back into the first equation (400 + x^2 = y^2) gives:

y = 43.67424242...

And x + y = 82.5 feet.

A quick check with the law of sines shows that these values work for both triangles. That's a big pole!

Posted by Jyqm on 2006-05-01 14:15:59