Question

Suppose you see a rock climber on a rock wall at a 32 degree angle of elevation. You are 1000 feet from the base of the rock wall. What is the height from the rock climber to the ground?

Answer 1

To find the height of the rock climber above the ground, we can use the tangent function from trigonometry. The tangent of an angle in a right triangle is equal to the opposite side (the height we're looking for) divided by the adjacent side (the distance from the base).

Given:

- Angle of elevation (\( \theta \)) = 32 degrees
- Distance from the base of the wall (adjacent side) = 1000 feet

We can use the formula:

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

Rearranging this gives us:

\[
\text{opposite} = \tan(\theta) \times \text{adjacent}
\]

Substituting the known values:

\[
\text{height} = \tan(32^\circ) \times 1000
\]

Using a calculator to find \( \tan(32^\circ) \):

\[
\tan(32^\circ) \approx 0.6249
\]

Now, calculate the height:

\[
\text{height} \approx 0.6249 \times 1000 \approx 624.9 \text{ feet}
\]

Therefore, the height from the rock climber to the ground is approximately **624.9 feet**.