To find the horizontal distance from the base of the tower to where the zip line ends, we can use the properties of a right triangle. The vertical height of the tower is the opposite side to the angle of elevation, the horizontal distance is the adjacent side, and the length of the zip line is the hypotenuse.
Given:
- Height (opposite side) = 120 feet
- Angle of elevation = 65 degrees
We can use the tangent function, which relates the angle to the opposite and adjacent sides:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
Rearranging this equation to solve for the adjacent side (horizontal distance \( d \)) gives us:
\[ d = \frac{\text{opposite}}{\tan(\theta)} \]
Substituting the known values:
\[ d = \frac{120 \text{ feet}}{\tan(65^\circ)} \]
Now we compute the tangent of \( 65^\circ \):
\[ \tan(65^\circ) \approx 2.1445 \]
Now substituting that value back into the equation:
\[ d \approx \frac{120}{2.1445} \approx 55.9 \text{ feet} \]
Thus, the horizontal distance from the base of the tower to where the zip line ends is approximately 55.9 feet.
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