Question

A surveying team set up some equipment 104104 ft from the base of a tree in order to sight the top of the tree. From ground level, they measure the angle of elevation to be 27.5∘27.5∘. If the calculated height needs to be accurate to within 1.51.5%, what is the allowed error in the angle measurement? (Assume the 104104 ft measurement is 100% accurate.)

Answer 1

Answer:

Relative error = 0.0254 %

Step-by-step explanation:

Given details

Height of equipment from base is 104 ft

Elevation angle is 27.5 degree

From figure we have

$tan\theta = \frac{x}{104}$ .........1

$tan(27.5) = \frac{x}{104}$

x  = 54.13 ft

differentiating 1st eq wrt x

$\frac{d}{dx} tan\theta = \frac{d}{dx} [\frac{x}{104}]$

$sec^2\theta \frac{d\theta}{dx} = \frac{1}{104}$

Taken $d\theta \ as \Delta x \ and\ dx = \Delta x$ for minute change

$\Delta x = 1.5% x$

             $= 0.015 \times 54.13 = 0.8119 ft$

from $sec^2 \theta \frac{d\theta}{dx} = \frac{1}{104}$

     $\Delta \theta = \frac{1}{104 sec^2 \theta} \Delta x$

$\Delta \theta = \frac{1}{104 \times 1.115} 0.8119$

$\Delta \theta = 0.007$

Reltaive error$= \frac{\Delta \theta}{\theta} \times 100$

                      $= \frac{0.007}{27.5}\times 100$

Relative error = 0.0254 %