Question

A balloon is directed over a line between two observers who are 1km apart. The angle of elevation of the balloon's gondola from one observer is 22°, from the other, it is 30°. What is the altitude of the gondola?

Answer 1

Not much calculus here -- just trig.
If the point below the balloon is x meters from the 22° observer, then if its height is h, you have

h/x = tan22°
h/(1000-x) = tan30°

That means

h/(1000-h/tan22°) = tan30°
or, after a little algebra,
h = 1000tan30°/(1+tan30°/tan22°)