Calculating the Height of a Kite using Angle of Elevation
<p>Let \( h \) be the height of the kite above the ground and \( d \) be the horizontal distance between observer \( P \) and the point on the ground directly below the kite.</p>
<p>For observer \( P \), using the tangent of the elevation angle:</p>
<p>\[ \tan(38^\circ) = \frac{h}{d} \rightarrow d = \frac{h}{\tan(38^\circ)} \]</p>
<p>For observer \( Q \), \( Q \) is 15 m apart from \( P \) on the horizontal plane, so the horizontal distance between \( Q \) and the point on the ground directly below the kite is \( d + 15 \) m.</p>
<p>Using the tangent of the elevation angle for \( Q \):</p>
<p>\[ \tan(45^\circ) = \frac{h}{d + 15} \]</p>
<p>Since \( \tan(45^\circ) = 1 \), we have:</p>
<p>\[ 1 = \frac{h}{d + 15} \rightarrow d + 15 = h \]</p>
<p>Substitute \( d \) from the first equation:</p>
<p>\[ \frac{h}{\tan(38^\circ)} + 15 = h \]</p>
<p>Solve for \( h \):</p>
<p>\[ h(\tan(38^\circ)) = h \tan(38^\circ) \]</p>
<p>\[ h = 15 \tan(38^\circ) \]</p>
<p>Calculate \( h \) using a calculator:</p>
<p>\[ h \approx 15 \times 0.7813 \]</p>
<p>\[ h \approx 11.7195 \]</p>
<p>The height of the kite to the nearest meter is approximately 12 meters.</p>
