Question

You're hiking in Zion National Park when you stop to look at a specific rock feature. The rock feature is 200 feet above the base of the mountain, and the diagonal distance between you and the rock feature is 850 feet. Using the inverse of sine, what is the angle of elevation you are using to look up at the feature? Round your answer to the nearest whole degree.

Answer 1

To find the angle of elevation, we can use the definition of sine in a right triangle. The sine of an angle is equal to the length of the opposite side divided by the length of the hypotenuse.

In this scenario:
- The height of the rock feature above the base of the mountain is the opposite side, which is 200 feet.
- The diagonal distance from your position to the rock feature is the hypotenuse, which is 850 feet.
- The angle of elevation \( \theta \) is what we are trying to find.

We can set up the equation using the sine function:

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{200}{850}
\]

Now, we can calculate \( \frac{200}{850} \):

\[
\frac{200}{850} = \frac{2}{8.5} \approx 0.2353
\]

Next, we use the inverse sine function (arcsin) to find the angle \( \theta \):

\[
\theta = \arcsin(0.2353)
\]

Using a calculator to compute this:

\[
\theta \approx 13.5^\circ
\]

Rounding to the nearest whole degree gives:

\[
\theta \approx 14^\circ
\]

Thus, the angle of elevation is approximately \( \boxed{14} \) degrees.