Two poles, 60 feet tall and 20 feet tall, stand on opposite sides of a field. The poles are 80 feet apart. Support cables are placed from the top of each pole to the bottom of the opposite pole. How far above the ground is the intersection of the cables. What if the poles were 120 feet a part.
The height h is given by 1/h = 1/60 + 1/20
h = 15'
This is true regardless of the width of the field.
1--Let A and B = the two height of the two poles.
2--Let X = the height of the poles crossing.
3--Let C be the distance between the two poles.
4--From the figure, A/C = X/Y or AY = CX.
5--Similarly, B/C = X/(C - Y) or BY = BC - CX.
6--Y = CX/A = (BC - CX)/B from which X = AB/(A+B).
Note - X is actually one half the harmonic mean of the two dimensions A and B, the harmonic mean being 2AB/(A + B).
Therefore, the height of the crossing is totally independant of the distance between the two buildings.
what is y?
To find the height at which the support cables intersect, we can use similar triangles.
Let's start with the case where the poles are 80 feet apart.
In this case, we have two similar triangles: one formed by the taller pole, the distance between the poles, and the height of the intersection point, and another formed by the shorter pole, the distance between the poles, and the height of the intersection point.
The ratio of corresponding sides in similar triangles is equal. So, the ratio of the height of the taller pole to the distance between the poles is equal to the ratio of the height of the intersection point to the distance between the poles.
Let's solve for the height of the intersection point:
(height of taller pole) / (distance between poles) = (height of intersection point) / (distance between poles)
60 / 80 = (height of intersection point) / 80
Cross multiplying, we get:
80 * 60 = 80 * (height of intersection point)
4800 = 80 * (height of intersection point)
Dividing both sides by 80, we find:
(height of intersection point) = 4800 / 80 = 60
Therefore, the height at which the support cables intersect above the ground is 60 feet when the poles are 80 feet apart.
Now let's consider the case where the poles are 120 feet apart.
Using the same method as above, we can solve for the height of the intersection point:
(height of taller pole) / (distance between poles) = (height of intersection point) / (distance between poles)
60 / 120 = (height of intersection point) / 120
Cross multiplying:
120 * 60 = 120 * (height of intersection point)
7200 = 120 * (height of intersection point)
Dividing both sides by 120:
(height of intersection point) = 7200 / 120 = 60
Again, the height at which the support cables intersect above the ground is 60 feet when the poles are 120 feet apart.
Therefore, the height at which the support cables intersect does not change with the distance between the poles.