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      <text>18.4.2.1 Measuring the height of a building</text>
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<h3><span>Engineering example</span></h3>The height \(h\) of a building is estimated from
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    <li>the known horizontal distance \(x\) between the point of observation \(M\) and the foot of the building and</li>
    <li>the elevation angle \(\theta\) between the horizontal and the line joining the point of observation to the top of the building (see Figure below).
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<p>If the measured horizontal distance is \(x = 150 \text{m}\) and the elevation angle is \(\theta = 40^{\circ}\), estimate the error in measured building height due to an error of \(0.1^{\circ}\) degree in the measurement of the angle of elevation.</p>
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<p>The variables \(x, \theta\), and \(h\) are related by \[\tan \theta = \dfrac{h}{x}\] or \[x \tan \theta = h.\]</p>
<p>The error in \(h\) resulting from a measurement error in \(\theta\) can be deduced by differentiating \(x \tan \theta = h\) as follows. \[\dfrac{\mathrm{d}(x \tan \theta)}{\mathrm{d} \theta} = \dfrac{\mathrm{d} h}{\mathrm{d} \theta} \quad \implies \quad
    \tan \theta \dfrac{\mathrm{d} x}{\mathrm{d} \theta} + x \dfrac{\mathrm{d}(\tan \theta)}{\mathrm{d} \theta} = \dfrac{\mathrm{d} h}{\mathrm{d} \theta}.\] This can be written \[\tan \theta \dfrac{\mathrm{d} x}{\mathrm{d} \theta} + x \sec^2 \theta = \dfrac{\mathrm{d}
    h}{\mathrm{d} \theta}.\]</p>
<p>The equation above gives the relationship among the small variations in variables \(x, h\) and \(\theta\). Since \(x\) is assumed to be without error and independent of \(\theta\), \(\dfrac{\mathrm{d}x}{\mathrm{d} \theta} = 0\) and the equation becomes
    \[x \sec^2 \theta = \dfrac{\mathrm{d}h}{\mathrm{d} \theta}.\]</p>
<p>The above equation can be considered to relate the error in building height \(\delta h\) to the error in angle \(\delta \theta\): \[\dfrac{\delta h}{\delta \theta} \simeq x \sec^2 \theta.\]</p>
<p>It is given that \(x = 150 \text{m}\).</p>
<p>The incidence angle \(\theta = 40^{\circ}\) can be converted to radians i.e \(\theta = \dfrac{40 \pi}{180} \text{ rad} = \dfrac{2 \pi}{9} \text{ rad}\).</p>
<p>Then the error in angle \(\delta \theta = 0.1^{\circ}\) needs to be expressed in radians for consistency of the units.</p>
<p>So \(\delta \theta = \dfrac{0.1 \pi}{180} \text{ rad} = \dfrac{\pi}{1800} \text{ rad}\). Hence, \[\delta h = 150 \dfrac{\pi}{1800 \times \cos^2\left(\dfrac{2\pi}{9}\right)} \approx 0.45 \text{ m}. \] </p>
<p>So the error in building height resulting from an error in elevation angle of \(0.1^{\circ}\) is about \(0.45 \text{ m}\).</p>]]></text>
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