Distance? (22)?

Question

A hot-air baloon is held at a constant altitude by two ropes that are anchored to the ground. One rope is 120 feet long and makes an angle of 65 degrees with the ground. The other rope is 115 feet long. What is the distance between the points on the ground at which the two ropes are anchored?

Answer 1

You have a triangle formed by the two ropes and the distance on the ground between where they are anchored.

Let
x = distance between points on ground where ropes are anchored
65° = angle opposite 115 foot rope
θ = angle opposite 120 foot rope
φ = angle opposite ground

By the Law of Sines

115/sin 65° = 120/sinθ

sinθ = (120/115)sin65°
θ = arcsin[(120/115)sin65°] ≈ 71.034153°

φ = 180° - 65° - θ = 115° - θ ≈ 43.965847°
sinφ ≈ 0.6942294

By the Law of Sines

x/sinφ = 115/sin 65°
x = 115(sinφ / sin 65°) ≈ 88.089708 ft