linearization and differentials: estimating height of a building

**PROBLEM:**

A surveyor, standing 30 ft 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 error in estimating the height of the building to be less than 4%?

**ATTEMPT:**

I'm not really sure how to approach this problem. The condition requires that the estimation and the true value be less than 4%; or,

$\displaystyle \frac{df}{f(x)}*100<4$

where df and f(x) represents the height of the building and its differential is a result of the propagation of error in measuring the angle of elevation.

Re: linearization and differentials: estimating height of a building

Quote:

Originally Posted by

**Lambin** **PROBLEM:**

A surveyor, standing 30 ft 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 error in estimating the height of the building to be less than 4%?

**ATTEMPT:**

I'm not really sure how to approach this problem. The condition requires that the estimation and the true value be less than 4%; or,

$\displaystyle \frac{df}{f(x)}*100<4$

where df and f(x) represents the height of the building and its differential is a result of the propagation of error in measuring the angle of elevation.

Hi Lambin! :)

Suppose x is the angle and f(x) is the height as function of the angle.

Can you say what the formula for f(x) is?

The propagation of an error in the angle of $\displaystyle \Delta x$ is given by:

[1] $\displaystyle \Delta f = f'(x) \Delta x$

The percentage error in f is:

[2] $\displaystyle {\Delta f \over f(x)} \times 100\% < 4\%$

Can you substitute [1] in [2] and find $\displaystyle \Delta x$?