Question
If a tree has a height of 47 feet, what would be the angle of elevation from level ground measeured from 42 feet away? Round your answer to the nearest tenth of a degree.
Answers
GPT-4o mini
To find the angle of elevation from a point on level ground to the top of a tree, we can use the tangent function from trigonometry. The formula is:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
In this scenario:
- The height of the tree (opposite side) is 47 feet.
- The distance from the point on the ground to the base of the tree (adjacent side) is 42 feet.
We can set up the equation:
\[
\tan(\theta) = \frac{47}{42}
\]
To find the angle \(\theta\), we take the arctangent (inverse tangent) of both sides:
\[
\theta = \tan^{-1}\left(\frac{47}{42}\right)
\]
Using a calculator:
1. Calculate \(\frac{47}{42} \approx 1.1190\).
2. Now compute the arctangent:
\[
\theta = \tan^{-1}(1.1190) \approx 48.37^\circ
\]
Rounding this to the nearest tenth of a degree gives:
\[
\theta \approx 48.4^\circ
\]
Therefore, the angle of elevation is approximately \(48.4^\circ\).
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
In this scenario:
- The height of the tree (opposite side) is 47 feet.
- The distance from the point on the ground to the base of the tree (adjacent side) is 42 feet.
We can set up the equation:
\[
\tan(\theta) = \frac{47}{42}
\]
To find the angle \(\theta\), we take the arctangent (inverse tangent) of both sides:
\[
\theta = \tan^{-1}\left(\frac{47}{42}\right)
\]
Using a calculator:
1. Calculate \(\frac{47}{42} \approx 1.1190\).
2. Now compute the arctangent:
\[
\theta = \tan^{-1}(1.1190) \approx 48.37^\circ
\]
Rounding this to the nearest tenth of a degree gives:
\[
\theta \approx 48.4^\circ
\]
Therefore, the angle of elevation is approximately \(48.4^\circ\).