Question

You’re hiking in Zion National Park when you stop to look at a specific rock feature. The rock feature is 300 feet above the base of the mountain, and the diagonal distance between you and the rock feature is 750 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.(1 point)

Answer 1

To find the angle of elevation using the inverse sine, we can use the relationship between the height of the rock feature, the distance from the observer to the base of the mountain, and the angle of elevation.

Let:
- \( h = 300 \) feet (the height of the rock feature above the base of the mountain)
- \( d = 750 \) feet (the diagonal distance from the observer to the rock feature)

We will first calculate the horizontal distance from the observer to the base of the mountain using the Pythagorean theorem, which states:

\[
d^2 = h^2 + b^2
\]

where \( b \) is the horizontal distance. Rearranging gives us:

\[
b^2 = d^2 - h^2
\]
\[
b = \sqrt{d^2 - h^2}
\]

Now substituting the known values:

\[
b = \sqrt{750^2 - 300^2}
\]
\[
b = \sqrt{562500 - 90000}
\]
\[
b = \sqrt{472500}
\]
\[
b \approx 688.11 \text{ feet}
\]

Next, we can find the angle of elevation \( \theta \) using the sine function:

\[
\sin(\theta) = \frac{h}{d}
\]
\[
\sin(\theta) = \frac{300}{750}
\]
\[
\sin(\theta) = 0.4
\]

Now we find \( \theta \) using the inverse sine function:

\[
\theta = \arcsin(0.4)
\]

Calculating this gives:

\[
\theta \approx 23.58 \text{ degrees}
\]

Rounding to the nearest whole degree, the angle of elevation is:

\[
\theta \approx 24 \text{ degrees}
\]

Thus, the angle of elevation is **24 degrees**.