To solve this problem, follow these steps:
1. Understand the given data:
- Horizontal distance from the tree: [tex]\( 20\pi \)[/tex] meters.
- Height of the observer above the ground: 1.5 meters.
- Angle of elevation to the top of the tree: [tex]\( 30^\circ \)[/tex].
2. Convert the angle from degrees to radians:
[tex]\[
\text{Angle in radians} = 30^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{6} \approx 0.524 \text{ radians}
\][/tex]
3. Apply trigonometry to find the height of the tree:
- The tangent of the elevation angle ([tex]\( \theta \)[/tex]) is given by:
[tex]\[
\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}
\][/tex]
In this context:
- [tex]\(\theta = 30^\circ\)[/tex]
- Opposite = height of the tree above the observer's eyes.
- Adjacent = horizontal distance from the tree = [tex]\( 20\pi \)[/tex].
- Rearrange the trigonometric formula to solve for the "Opposite" side:
[tex]\[
\text{Opposite} = \tan(30^\circ) \times 20\pi
\][/tex]
4. Calculate the tangent of 30 degrees:
[tex]\[
\tan(30^\circ) = \frac{1}{\sqrt{3}} \approx 0.577
\][/tex]
5. Compute the height above the observer's eyes:
[tex]\[
\text{Height above the observer's eyes} = 0.577 \times 20\pi \approx 36.276 \text{ meters}
\][/tex]
6. Add the observer's height to the computed height:
- Observer's height from the ground: 1.5 meters
- Total height of the tree:
[tex]\[
\text{Tree height} = 36.276 + 1.5 \approx 37.776 \text{ meters}
\][/tex]
7. Round the height to the nearest meter:
[tex]\[
\text{Tree height rounded} \approx 38 \text{ meters}
\][/tex]
Thus, the height of the tree to the nearest meter is 38 meters.
