Question

For the following exercises, find the area of each triangle. Round each answer to the nearest tenth.

A communications tower is located at the top of a steep hill, as shown in Figure 30. The angle of inclination of the hill is 67°. A guy wire is to be attached to the top of the tower and to the ground, 165 meters downhill from the base of the tower. The angle formed by the guy wire and the hill is 16°. Find the length of the cable required for the guy wire to the nearest whole meter.

Answer 1

Consider a communication tower is located at the top of a steep hill. The angle of inclination of the hill is 67°. A guy wire is attached to the top of the tower and to the ground, 165m downhill from the base of the tower. The angle formed by the guy wire and the hill is 16°.

 

The objective is to find the length of the cable required for the guy wire.

 

The sketch of the tower is:

 

Here AC represents the tower and CBD is the inclination of the hill

 

In ΔABD ∠ABD = ∠ABC + ∠CBD

 

Then,

∠ABD = ∠ABC + ∠CBD

           = 16° + 67°

           = 83°

 

Apply the sum of the angles property in ΔABD then, ∠A + ∠B + ∠D = 180°

 

Since ∠B = 83°, ∠D = 90°

then,

 ∠A + ∠B + ∠D = 180°

∠A + 83° + 90° = 180°

                    ∠A = 180° - 173°

                    ∠A = 7°

 

Apply the sum of the angles property in ΔABC then, ∠A + ∠B + ∠C = 180°.

 

Substitute ∠A = 70°, ∠B = 16° in ∠A + ∠B + ∠C = 180°.

Then,

∠A + ∠B + ∠C = 180°

  7° + 16° + ∠C = 180°

                    ∠C = 180° - 23°

                   ∠C = 157°

 

In ΔABC ∠A = 7°, BC = 165, ∠C = 157°.

 

Find AB.

Apply the law of sines then,

     AB/sin C = BC/sin A

AB/sin157° = 165/sin7°

               AB = 165/sin7° × sin157°

               AB = 167/0.12186 × 0.39073

               AB = 529.053

 

Hence, the length of the cable required for the guy wire is 529m.

Hence, the length of the cable required for the guy wire is 529m.