Question

A communications tower is located at the top of a steep hill, as shown. The angle of inclination of the hill is 58º. A guy wire is to be attached to the top of the tower and to the ground, 125 m downhill from the base of the tower. The angle a in the figure is determined to be 14°. Find the length of cable required for the guy wire. (Round your answer to the nearest meter.)

Answer 1

The length of cable required for the guy wire is of 214.4 meters.

Explanation:

Drawing of the situation:

The first step to solve this question is finding angles A and B.

To find these angles, we first find angle C.

The sum of the internal angles of a triangle is 180º. So

58 + 90 + C = 180

148 + C = 180

C = 32

Angles A and C are supplementary, that is, they add to 180. So

C + A = 180

32 + A = 180

A = 148º

Since the sum of the interior angles of a triangle is 180º.

A + B + 14º = 180º

148 + B + 14 = 180

B = 18º

Now, we have the following right triangle, to find the base:

In a right triangle, the sine of an angle is the length of the side opposite to the angle divided by the hypotenuse. So

sin(32º) = b/125

We have that the sine of 32º is 0.53. So

0.53 = b/125

b = 125*0.53 = 66.25

Now, we look at another right triangle, to find the length of the cable required.

The cosine of an angle is the length of the side adjacent to this angle divided by the hypotenuse. So

cos(72º) = 66.25/x

We have that the cosine of 72 degrees is 0.309. So

0.309 = 66.25/x

0.309x = 66.25

x = 66.25/0.309

x = 214.4

The length of cable required for the guy wire is of 214.4 meters.