Height of a hot air balloon using Law of Sines

Question

The angles of elevation of a balloon from two points A and B on level ground are 24.17 degrees and 47.67 degrees respectively. Points A and B are 8.4 miles apart, and the balloon is between the points, in the same vertical plane. Approximate the height of the balloon above the ground.

Answer 1

In the triangle ADC tanA = opposite/adjacent

Let opposite side = x, AB = AD + BD = 8.4, and angle A = 24.17, angle B = 47.67

The adjacent = 4.2

tanA = opposite/adjacent

tan(24.17) = x/4.2

The trigonometric table in tan(24.17) = 0.4487

0.4481 = x/4.2

Multiply each side by 4.2

(0.4481)(4.2) = x

x = 1.88202

Therefore the height of the balloon above the ground is 1.88 miles.