Question

A telephone pole has a wire attached to its top that is anchored to the ground. the distance from the bottom of the pole to the anchor point is 68 feet less than the height of the pole. if the wire is to be 4 feet longer than the height of the pole, what is the height of the pole?

Answer 1

The height of the pole is 96 feet

In mathematics, the Pythagorean theorem, or Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the legs a, b and the hypotenuse c, often called the Pythagorean equation:[1]

$a^{2}+b^{2}=c^{2},$

Create a diagram of the scenario first. You would have a right triangle with a hypotenuse (longest side) of h + 4, a longest leg of h - 68, and one leg of length h to represent the pole.

Set up the equation $h^2 + (h - 68)^2 = (h + 4)^2$ using the Pythagorean theorem ($a^2 + b^2 = c^2$).

$h^2 + (h - 68)^2 = (h + 4)^2$

On simplifying we get

$h^2+h^2-136h+4624 = h^2+8h+16$

$2h^2-136h+4624=h^2+8h+16$

$h^2-144h+4608=0$

Solving using quadratic formula

$h_{1,\:2}=\frac{-\left(-144\right)\pm \sqrt{\left(-144\right)^2-4\cdot \:1\cdot \:4608}}{2\cdot \:1}$  

$h_1=\frac{-\left(-144\right)+48}{2\cdot \:1},\:h_2=\frac{-\left(-144\right)-48}{2\cdot \:1}$

h1 = 96 feet , h2 = 48 feet

If height would have been 48 feet then the other side would have a negative value as as 48 - 68 = -20 .

Hence the height of the pole is 96 feet

Learn more about Pythagoras theorem here :

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