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.
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