Question
If a tree has a height of 126 feet, what would be the angle of elevation from level ground measured from 79 feet away? Round your answer to the nearest tenth of a degree.
Answers
GPT-4o mini
To find the angle of elevation from the ground to the top of the tree, we can use the tangent function from trigonometry. The height of the tree is the opposite side of the triangle, and the distance from the tree is the adjacent side.
Let:
- \( h = 126 \) feet (height of the tree)
- \( d = 79 \) feet (distance from the tree)
The angle of elevation \( \theta \) can be found using the tangent function:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d} = \frac{126}{79}
\]
Now, calculate \( \tan(\theta) \):
\[
\tan(\theta) = \frac{126}{79} \approx 1.5975
\]
Next, we find the angle \( \theta \) by taking the arctangent (inverse tangent) of \( 1.5975 \):
\[
\theta = \tan^{-1}(1.5975)
\]
Using a calculator to find \( \tan^{-1}(1.5975) \):
\[
\theta \approx 57.7^\circ
\]
Therefore, the angle of elevation from the ground to the top of the tree, rounded to the nearest tenth of a degree, is:
\[
\boxed{57.7}
\] degrees.
Let:
- \( h = 126 \) feet (height of the tree)
- \( d = 79 \) feet (distance from the tree)
The angle of elevation \( \theta \) can be found using the tangent function:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d} = \frac{126}{79}
\]
Now, calculate \( \tan(\theta) \):
\[
\tan(\theta) = \frac{126}{79} \approx 1.5975
\]
Next, we find the angle \( \theta \) by taking the arctangent (inverse tangent) of \( 1.5975 \):
\[
\theta = \tan^{-1}(1.5975)
\]
Using a calculator to find \( \tan^{-1}(1.5975) \):
\[
\theta \approx 57.7^\circ
\]
Therefore, the angle of elevation from the ground to the top of the tree, rounded to the nearest tenth of a degree, is:
\[
\boxed{57.7}
\] degrees.