Expert: Ahmed Salami - 2/16/2009

Question
Two observers are 500 feet apart. Each measures the angle of elevation of a hot air balloon that lies in a vertical plane passing through their locations; the angles are 53 and 42. Find the height of the balloon if the observers are on opposite sides of the balloon. (Simplify the answer as much as possible.)

Answer
Hi Holly,
I've provided no diagram here and so you would have to visualize the situation.
Let the balloon be a height h and at a horizontal distance x from the observer with the angle of elevation 53 degrees, it would then be at a horizontal distance (500-x)from the observer with the angle of elevation 42 degrees.

From the first observer,
tan53 = h/x
h = x(tan53)
h = 1.327x
and from the second,
tan42 = h/(500-x)
h = (500-x)(tan42)
h = (500-x)(0.9)

Equating,
1.327x = (500-x)(0.9)
1.327x = 450 - 0.9x
1.327x + 0.9x = 450
2.227x = 450
x = 202.16

But,
h = 1.327x
 = 1.327(202.16)
 = 268.26 feet

Regards