Question

A telephone pole has a wire to its top that is anchored to the ground. The distance from the bottom of the pole to the anchor point is 49 feet less than the height of the pole. If the wire is to be 14 feet longer than the height of the pole, what is the height of the pole?

Answer 1

The height of the pole is 105ft,

Explanation:

Let us call
`h` the height of the pole, then we know that the distance from the bottom of the pole to the anchor point is 49, or it is
`h - 49`.

The wire length
`d` is 14 ft longer than height
`h`, hence

`d= h+14`.

Thus we get a right triangle with hypotenuse
`d= y+14`, perpendicular
`h`, and base
`h-49`; therefore, the Pythagorean theorem gives

`(h-49)^2+h^2 = (h+14)^2`

which upon expanding we get:

`h^2-98h+2401 = h^2+28h+196`

further simplification gives

`h^2-126h+2205=0`,

which is a quadratic equation with solutions

`h =21ft\\h = 105ft.`

Since the first solution
`h =21ft` will give the triangle base length of
`21ft-49ft = -28ft` which is negative; therefore, we disregard it and pick the solution
`h = 105ft`.

Hence, the height of the pole is 105ft.