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.
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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".
