Question

A sightseeing balloon is sighted simultaneously by two people on the ground, 7 miles apart from each other. both people are directly east of the balloon. the angles of elevation are reported as 23^\circ and 59^\circ. how high is the balloon?

Answer 1

Answer:

about 3.9886 miles, or 21,060 ft

Step-by-step explanation:

The mnemonic SOH CAH TOA reminds us of the relationship between sides of a right triangle and the trig functions of the angles. If the balloon height is represented by h, and the distance from the spot below the balloon to the nearest observer is x, then we know ...

tan(59°) = h/x

and

tan(23°) = h/(x+7)

If we invert the ratios, we can express these a little differently:

cot(59°) = tan(31°) = x/h

cot(23°) = tan(67°) = (x+7)/h

We can multiply both equations by h and subtract the first product from the second.

h·tan(67°) -h·tan(31°) = (x+7) -(x)

h(tan(67°) -tan(31°)) = 7

h = 7/(tan(67°) -tan(31°)) ≈ 3.988623 . . . . miles

The height of the balloon is about 3.9886 miles.

_____

Comment on this geometry

The working of this problem assumes a flat earth. The altitude is such that oxygen would be required by the "sightseers".