Question - Calculating the Height of a Kite using Angle of Elevation

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.

For observer \( P \), using the tangent of the elevation angle:

\[ \tan(38^\circ) = \frac{h}{d} \rightarrow d = \frac{h}{\tan(38^\circ)} \]

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.

Using the tangent of the elevation angle for \( Q \):

\[ \tan(45^\circ) = \frac{h}{d + 15} \]

Since \( \tan(45^\circ) = 1 \), we have:

\[ 1 = \frac{h}{d + 15} \rightarrow d + 15 = h \]

Substitute \( d \) from the first equation:

\[ \frac{h}{\tan(38^\circ)} + 15 = h \]

Solve for \( h \):

\[ h(\tan(38^\circ)) = h \tan(38^\circ) \]

\[ h = 15 \tan(38^\circ) \]

Calculate \( h \) using a calculator:

\[ h \approx 15 \times 0.7813 \]

\[ h \approx 11.7195 \]

The height of the kite to the nearest meter is approximately 12 meters.