Question

A communications tower is located at the top of a steep hill, as shown. The angle of inclination of the hill is 68°. A guy wire is to be attached to the top of the tower and to the ground, 148 m downhill from the base of the tower. The angle formed by the guy wire is 9°. Find the length of the cable required for the guy wire.

Answer 1

The length of the cable required for the guy wire is approximately 158.4 meters.

Step-by-step explanation:

Identify the relevant information:

Angle of inclination of the hill (θ) = 68°

Distance from the base of the tower to the guy wire attachment point (d) = 148 m

Angle formed by the guy wire with the horizontal (α) = 9°

Visualize the situation:

Create a rough sketch of the situation, including the tower, the hill, the guy wire, and the angles.

Formulate the equations:

We need to find the length of the guy wire (L). We can use trigonometry to relate the given information to L:

Vertical component of the hill:

h = d * sin(θ)

Horizontal component of the hill:

x = d * cos(θ)

Triangle formed by the guy wire:

sin(α) = x / L

cos(α) = h / L

Solve for the length of the cable:

Square both equations in the triangle:

sin^2(α) = x^2 / L^2

cos^2(α) = h^2 / L^2

Add the two equations:

sin^2(α) + cos^2(α) = (x^2 + h^2) / L^2

Substitute the trigonometry identity sin^2(α) + cos^2(α) = 1:

1 = (x^2 + h^2) / L^2

Solve for L:

L = sqrt(x^2 + h^2)

Substitute the values:

h = d * sin(θ) = 148 m * sin(68°) ≈ 142.2 m

x = d * cos(θ) = 148 m * cos(68°) ≈ 43.4 m

L = sqrt(x^2 + h^2) ≈ sqrt(43.4^2 + 142.2^2) ≈ 158.4 m

Therefore, the length of the cable required for the guy wire is approximately 158.4 meters.