To solve this problem, we can use some trigonometry. We have a right triangle formed by the height of the zipline (the vertical leg), the horizontal distance (the horizontal leg), and the hypotenuse (the zipline itself).
The height of the zipline is given as \( 120 \) feet, and we know the angle of elevation from the ground to the zipline is \( 65^\circ \).
We can use the tangent function, which is defined as the ratio of the opposite side to the adjacent side in a right triangle:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
In our case:
- The opposite side is the height of the zipline, which is \( 120 \) feet.
- The adjacent side is the horizontal distance \( d \) from the base of the tower to where the zipline ends.
Setting up the equation:
\[ \tan(65^\circ) = \frac{120}{d} \]
Now we can solve for \( d \):
\[ d = \frac{120}{\tan(65^\circ)} \]
Now calculating \( \tan(65^\circ) \):
Using a calculator, we find:
\[ \tan(65^\circ) \approx 2.1445 \]
So:
\[ d = \frac{120}{2.1445} \approx 55.9 \text{ feet} \]
Thus, the horizontal distance from the base of the tower to where the zipline ends is approximately \( 55.9 \) feet.
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