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 %

Answer 2

Relative error is simply the ratio of the absolute error and actual value.

The relative error is 0.02236%

The given parameters are:

$\mathbf{Height = 104}$

$\mathbf{\theta = 27.5}$

The relationship between the height, and the base (x) of a triangle is:

$\mathbf{tan(\theta) = \frac{x}{104}}$

So, we have:

$\mathbf{x = 104tan(\theta)}$

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

$\mathbf{x = 54.14}$

Recall that: $\mathbf{tan(\theta) = \frac{x}{104}}$

Differentiate

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

The change in the base (x) is:

$\mathbf{\Delta x = 1.5\% \times x}$

$\mathbf{\Delta x = 1.5\% \times 54.14}$

$\mathbf{\Delta x = 0.8121}$

Recall that:

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

So, we have:

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

Rewrite as:

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

Substitute known values

$\mathbf{ \Delta \theta = \frac{1}{104sec^2(30)}\times 0.8121}$

$\mathbf{ \Delta \theta = \frac{1}{104 \times 1.2701 }\times 0.8121}$

$\mathbf{ \Delta \theta = 0.00615}$

So, the relative error is:

$\mathbf{Error = \frac{ \Delta \theta }{\theta } \times 100\%}$

$\mathbf{Error = \frac{ 0.615}{27.5}\%}$

$\mathbf{Error = 0.02236\%}$

Hence, the relative error is 0.02236%

Read more about relative errors at:

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