Question

A cell phone tower is anchored by two cables on each side for support. The cables stretch from the top of the tower to the ground, with each being equidistant from the base of the tower. The angle of depression from the top of the tower to the point in which the cable reaches the ground is 23°. If the tower is 140 feet tall, find the ground distance between the cables.

Answer 1

660 ft (rounded to the nearest foot)

Explanation:

The mnemonic SOH CAH TOA reminds you that the relation between the sides of a right triangle and one of the acute angles is ...

Tan = Opposite/Adjacent

Here, we have the angle between the tower and the cable being (90°-23°) = 67°, so the distance from one cable to the tower (d) is ...

tan(67°) = d/(140 ft)

Then the distance between two cable anchor points on opposite sides of the tower will be ...

2d = 2·tan(67°)·(140 ft) = 659.64 ft

The ground distance between cables is about 660 ft.

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We have assumed that the wording "two cables on each side" means "two cables, one on each side". While the angle of depression is unusually small, we have assumed it to be as given.