Question

If a tree has a height of 31 feet, what would be the angle of elevation from level ground measured from 38 feet away? Round your answer to the nearest tenth of a degree.

Answer 1

To determine the angle of elevation from a point on level ground to the top of a tree, we can use trigonometry, specifically the tangent function.

We can denote:
- the height of the tree (\( h \)) as 31 feet,
- the distance from the point on the ground to the base of the tree (\( d \)) as 38 feet,
- the angle of elevation (\( \theta \)) that we want to find.

According to the tangent function, we have:

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

Substituting in the values we have:

\[
\tan(\theta) = \frac{31}{38}
\]

Now we calculate \( \tan^{-1} \left(\frac{31}{38}\right) \) to find the angle \( \theta \):

First, calculate \( \frac{31}{38} \):

\[
\frac{31}{38} \approx 0.815789
\]

Now find \( \theta \):

\[
\theta = \tan^{-1}(0.815789)
\]

Using a calculator to compute \( \tan^{-1}(0.815789) \):

\[
\theta \approx 39.5^\circ
\]

Finally, rounding to the nearest tenth of a degree, we find that the angle of elevation from level ground to the top of the tree is approximately:

\[
\boxed{39.5}^\circ
\]