A crew is setting up a zip line between two towers with support wires as shown in the figure (not to scale). they only know the heights of the towers and the angles that two of the supports make with the ground. what is the length of the zip line?

length of the zip line = 121. 63 ft

Step-by-step explanation:

The length of the zip line AB forms a triangle  ABC. To find the length AB we need to know the length of 2 sides of the triangle an angle.

Triangle ADC

We need to find the hypotenuse side AC using the SOHCAHTOA principle.

Therefore,

sin 41° = opposite/hypotenuse

opposite = 65 ft

sin 41° = 65/AC

cross multiply

0.65605902899 AC = 65

divide both sides by 0.65605902899

AC = 65/0.65605902899

AC = 99.0764506359  ft

AC ≈ 99.08 ft

Triangle BCE

We are looking for side BC. The triangle BCE is also a right angle triangle so we use the same methodology like the triangle ADC.

sin 62° = opposite/hypotenuse

opposite = 85 ft

sin 62° = 85/BC

cross multiply

BC sin 62° = 85

BC = 85/sin 62°

BC = 85/0.88294759285

BC = 96.2684543086

BC ≈ 96.27 ft

The angle ACB can be gotten when you subtract 62° and 41° from 180 (angle on a straight line).

Therefore,

∠ACB = 180° - 62° - 41°

∠ACB  = 77°

Now let us use the cosine law to find the zip line AB.

c² = a² + b² - 2ab cos C

a = 96.27 ft

b = 99.08 ft

c² = 96.27² + 99.08² - 2 × 96.27 × 99.08 cos 77°

c² = 9267.9129 + 9816.8464 - 19076.8632 cos 77°  

c² = 19084.7593 - 19076.8632 × 0.22495105434

c²  = 19084.7593 - 4291.36049041

c² = 14793.398810

square root both sides

c = √14793.398810

c = 121.628116856

c ≈ 121. 63 ft

length of the zip line = 121. 63 ft