To solve this problem, we can use trigonometric functions. We have a right triangle formed by the vertical height of the zipline, the horizontal distance from the base of the tower to where the zipline ends, and the diagonal distance of the zipline itself.
Given:
- Height (opposite side) = 120 feet
- Angle of elevation = 65 degrees
We can use the tangent function, which is defined as the ratio of the opposite side to the adjacent side. In this case, we can let \( d \) represent the horizontal distance (adjacent side).
The tangent of the angle θ is given by:
\[ \tan(\theta) = \frac{{\text{opposite}}}{{\text{adjacent}}} \]
So, we have:
\[ \tan(65^\circ) = \frac{120}{d} \]
From this, we can rearrange to find \( d \):
\[ d = \frac{120}{\tan(65^\circ)} \]
Now we will calculate \( \tan(65^\circ) \) using a calculator:
\[ \tan(65^\circ) \approx 2.1445 \]
Now substituting back into the formula:
\[ d = \frac{120}{2.1445} \approx 55.9 \text{ feet} \]
Therefore, the horizontal distance from the base of the tower to where the zipline ends is approximately 55.9 feet.
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