If Two vertical poles have heights 6 ft and 12 ft a rope is stretched from the top of each pole to the bottom of the other how far above the ground do the ropes cross?

Answer 1

For this particular problem, the point of intersection is 4 feet above the ground.


The general formula for this situation is as follows:

Let h=height of the shorter pole
Let H=height of the taller pole

Then the height of the point of intersection, let's call it y, will be

y = (hH)/(h+H) or more symmetrically, 1/y = 1/h + 1/H

This may be deduced by simple algebra and the comparison of sides of similar triangles.

An interesting aspect of this problem is that the distance between the poles does not appear anywhere in the solution! No matter how far apart you stretch the poles, the height of the point of crossing remains the same - it just slides along a horizontal line at the calculated height.

Another point of interest is that the ratio of the horizontal distance from one pole to the crossing point ( call it x ) divided by the total distance between the two poles ( call it D ) also remains the same as the poles are moved apart. Specifically,

(x/D) = H/(h+H) = (a constant for a fixed set of poles)