The angles of depression of two points on the ground with respect to a hot-air balloon 4 miles up in the air are 13 degrees and 39 degrees. how far apart are the two points if they lie on a straight line that passes through the point on the ground that is directly below the balloon? round your answer to four decimal places.

26.2655 miles

Step-by-step explanation:

Whenever possible, draw a diagram.  I've attached a picture of the balloon and the lines of depression (not drawn to scale, of course).

As you can see, the angles of depression form alternate interior angles with the ground (angles of elevation).  So these angles are also 13° and 39°.

We can use the tangents of these angles to find the base lengths of the two triangles that are formed.

tan 13° = 4 / a

a = 4 / tan 13°

tan 39° = 4 / b

b = 4 / tan 39°

So the total distance is:

a + b

= 4 / tan 13° + 4 / tan 39°

≈ 22.2655