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Calculus/Percentage Error Word Problem

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Expert: - 7/26/2009


Question
A surveyor, standing 30 feet from the base of a building, measures the angle of elevation to the top of the building to be 75 degrees. How accurately must the angle be measured for the percentage (propagated) error in estimating the height of the building to be less then 4%?

I have drawn a picture to help me visualize the problem. I know I can figure out the actual size of the building through trig functions, but I am not sure how to calculate the percentage error to be less than 4%. Thanks for your time and any feedback you can might give me!

Answer
Draw a traingle with base 30, lower angle 75°, and height x, where the side of length x goes up and down and is perpendicular to the side of length 30.

This tan(75°) = x/30, or x = 30 tan(75°).

Since x is within 4%, x is between 0.96x and 1.04x.

This means the angle is between A and B, where 0.96x/30 = tanA and 1.04x/30 = tanB.

The value for x can be computed from the side of length 30 and the angle 75°.

This means that 0.96 tan(75°) = tanA, and A can be found from there.

It also means that 1.04 tan(75°) = tanB, and B can be found from here.

Once these two angles, A and B, aroud found, they can both be put in for y to get a percent.
(y-30)/3000.

Calculus

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