Question

If a tree has a height of 76 feet, what would be the angle of elevation from level ground measured from 29 feet away? Round your answer to the nearest tenth of a degree.

Answer 1

To find the angle of elevation from level ground to the top of the tree, we can use the tangent function in trigonometry.

The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side. In this case:

- The height of the tree is the opposite side (\(h = 76\) feet).
- The distance from the tree is the adjacent side (\(d = 29\) feet).

We use the following formula:

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d}
\]

Substituting the values we have:

\[
\tan(\theta) = \frac{76}{29}
\]

Calculating this gives:

\[
\tan(\theta) \approx 2.6207
\]

To find the angle \(\theta\), we take the arctangent (inverse tangent) of \(2.6207\):

\[
\theta = \tan^{-1}(2.6207)
\]

Using a calculator to find \(\theta\):

\[
\theta \approx 69.6 \text{ degrees}
\]

Rounding to the nearest tenth of a degree, the angle of elevation is:

\[
\boxed{69.6} \text{ degrees}
\]