Find the distance from the balloon to the soccer fields.

Question

A hot air balloon is flying above Groveburg. To the left side of the balloon, the balloonist measure the angle of depression to the Groveburg soccer fields to be twenty degrees 15’. To the right side of the balloon, the balloonist measures the angle of depression to the high school football field to be sixty two degrees 30’. The distance between the two athletic complexes is 4 miles.

Answer 1

First convert 20°15′ to decimal degrees:

20°15′ = 20.25°

Let hh be the height of the balloon. Let xx be the horizontal distance from the soccer fields to the point directly below the balloon.

Then the horizontal distance to the football field is 4−x4-x.

Using tangent:

tan(20.25°) = h/x

tan(62.5°) = h/(4-x)

So:

x tan(20.25°) = (4-x)tan(62.5°)

Solving gives:

x ≈ 3.356 miles

and

h ≈ 1.238 miles.

Now use the Pythagorean theorem to find the actual distance from the balloon to the soccer fields:

d = √(3.356² + 1.238²)

d ≈ 3.58 miles

So the balloon is about 3.58 miles from the soccer fields.