SOLUTION: A wire is attached to the top of a 36-foot pole. The other end of the wire is attached to the ground 20 feet from the bottom of the pole. If the pole makes an angle of 90degrees wi

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Question 276969: A wire is attached to the top of a 36-foot pole. The other end of the wire is attached to the ground 20 feet from the bottom of the pole. If the pole makes an angle of 90degrees with the ground, find the length of the wire to the nearest tenth of a foot.
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
Since the pole and the ground form a right triangle, and we know the length of 2 sides of this triangle, we can use the pythagorean formula to find the third side.

let a = 36 feet = length of the pole.

Let b = 20 feet = length from the bottom of the pole to the point where the wire is tied.

Let c = length from the point where the wire is tied to the top of the pole (this is the length of the wire)

pythagorean formula says:

a^2 + b^2 = c^2

This means that:

36^2 + 20^2 = c^2 which becomes:

c^2 = 1696

Take square root of both sides of this equation to get:

c = 41.181252056 feet.

We can also use trigonometry to arrive at the same answer.

angle opposite to a (the height of the pole) is the angle we are looking for.

We call it angle k.

This angle is also adjacent to the distance from the wire to the bottom of the pole. That would be side b.

The hypotenuse of the triangle is side c.

tangent (k) = opposite / adjacent = 36/20 = 1.8

Arctangent of 1.8 = 60.9453959 degrees.

Sine of 60.9453959 = opposite / hypotenuse = 36/c.

Multiply both sides by c and divide both sides by Sine of 60.9453959 and you get:

c = 36/Sine(60.9453959)

Solve for c to get:

c = 41.18252056 feet.

That's the same solution we got using the pythagorean formula.

Here's a rough sketch of the triangle formed: 


                             x

                             x x

                             x  x

                             x   x

                             x    x

                             x     x

                     a = 36  x      x    c = 41

                             x       x

                             x        x

                             x         x

                             x          x

                             x       (k) x

                             xxxxxxxxxxxxxx



                                b = 20