To find the height of point P above the level of point Q given the angle of elevation and the distance PQ, we can use trigonometry.
We have:
- The angle of elevation \(\theta = 40^\circ\)
- The distance \(PQ = 45\) km
We want to find the height \(h\) of point P above point Q. We can use the tangent of the angle:
\[ \tan(\theta) = \frac{\text{Opposite side (height)}}{\text{Adjacent side (distance)}} \]
Rearranging the formula to find the height \(h\):
\[ h = PQ \cdot \tan(\theta) \]
Now, substituting the values:
\[ h = 45 \cdot \tan(40^\circ) \]
Calculating \(\tan(40^\circ)\):
Using a calculator, \(\tan(40^\circ) \approx 0.8391\).
Now, calculate \(h\):
\[ h \approx 45 \cdot 0.8391 \approx 37.76 \text{ km} \]
Thus, the height of point P above the level of point Q is approximately \(37.76\) km.
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