To find the horizontal distance from the base of the tower to where the zip line ends, we can use trigonometry. Specifically, we can use the tangent function, since we have the height of the tower and the angle of elevation.
Given:
- Height of the tower (opposite side) = 120 feet
- Angle of elevation = 65°
The tangent of the angle of elevation is defined as the ratio of the opposite side to the adjacent side (horizontal distance).
\[ \tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} \]
Let \( d \) be the horizontal distance (adjacent side) from the base of the tower to where the zip line meets the ground.
We can rearrange the formula to solve for \( d \):
\[ d = \frac{\text{opposite}}{\tan(\text{angle})} \]
Substituting in the values we have:
\[ d = \frac{120 \text{ feet}}{\tan(65°)} \]
Now, calculating \( \tan(65°) \):
\[ \tan(65°) \approx 2.1445 \]
Now substitute this back into the equation for \( d \):
\[ d \approx \frac{120}{2.1445} \approx 55.9 \text{ feet} \]
Rounding to the nearest whole number, the horizontal distance \( d \) is approximately 56 feet.
Thus, the distance from the tower to where the zip line meets the ground is 56 feet.