Find the length of cable required for the guy wire. (Round your answer to the nearest meter.)

Question

A communications tower is located at the top of a steep hill, as shown. The angle of inclination of the hill is 58°. A guy wire is to be attached to the top of the tower and to the ground, 100 m downhill from the base of the tower. The angle 𝛼 in the figure is determined to be 7°.

Answer 1

Ashton, I guess the image that’s missing looks something like this:

T is the top of the tower, directly above H, the top of the hill.

C is the location of the guy wire stretching to T.

Extend the segment TH downward until it meets a horizontal line segment from C. Call this point of intersection M.

Distance CH is 100 m

Angle HCT is 7 degrees

What is the length of CT? Call it X.

How to solve it:

We now have two triangles with one equal (but unknown) side, CM. Call its length S

Now  use the definition of cosine once for each triangle:

Cos (58)= S / 100

Cos (58+7)= S / X

Now solve each for S

S = 100 cos(58)

S = X cos(65)

Equate the two Right Hand Sides

X cos(65) = 100 cos(58)

Now, I’ll let you solve for X, substituting the two cosines.