trigonometry problem?

Question

If h is the height of the balloon, and x is the distance along the ground from A to the point directly under the balloon, then the two equarions are
how could I arrive to the answer below? I have tried but it wouldn't work. please explain your way thoroughly. thanks
As a balloon passes between two points A and B, 2 miles apart, the angles of elevation of the balloon at these points are 27° 19' and 41° 45', respectively. Find the altitude of the balloon. Take A and B at the same level.

tan 27°19' = h/x, and
tan 41°45' = h/(2 – x)

You can solve this pair of equations for x and h.
Height = .654 miles = 3455 feet.

Answer 1

Are you having trouble getting to the two equations, or solving the two equations?

Getting to them is straightforward trigonometry.

Choosing x as the distance from A, the ground from A to x, the altitude from x up to the balloon, and the line of sight form a right triangle. The angle of elevation has as the opposite leg the altitude (h), and as the adjacent leg the distance to the point directly under the balloon (x). Thus, tan(a) = h/x

Since A and B are two miles apart, the distance from the balloon to B is 2-x. Again, the ground from B to the point under the balloon, the altitude up to the balloon from that point, and the line of sight form a right triangle. The angle of elevation has as the opposite leg the altitude (h), and as the adjacent leg the distance to the point directly under the balloon (2-x). Thus, tan(b) = h/(2-x).

====================
tan 27°19' = 0.516507 = h/x
tan 41°45' = 0.892534 = h/(2-x)

I'd just divide the first equation by the second, which would cancel out the h's:

0.516507 / 0.892534 = (h/x) / (h/(2-x))
0.578697 = (2-x)/x
0.578697x = 2-x
1.578697x = 2
x = 1.26686

Now that you have x, either equation can be used to solve h

0.516507 = h/x
0.516507 = h/1.26686
1.26686 * 0.516507 = h
0.65434 = h