Question

A zipline starts 120A zipline starts 120 feet above the ground and covers a diagonal distance, forming a triangle with the ground. If the angle of elevation where the zip line meets the ground is 65 degrees, what is the horizontal distance from the base of the tower to where the zip line ends?

Answer 1

To find the horizontal distance from the base of the tower to where the zip line ends, we can use some trigonometry. We have the following information:

- The height of the zip line above the ground (\( h \)) is 120 feet.
- The angle of elevation (\( \theta \)) is 65 degrees.

We can use the tangent function, which relates the angle of elevation to the opposite side (height of the tower) and the adjacent side (horizontal distance). The formula for tangent is:

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d}
\]

Where:
- \( h = 120 \) feet (the height of the zip line)
- \( d \) is the horizontal distance we are trying to find.

Rearranging this formula to solve for \( d \):

\[
d = \frac{h}{\tan(\theta)}
\]

Now, substituting in the known values:

\[
d = \frac{120}{\tan(65^\circ)}
\]

Using a calculator to find \( \tan(65^\circ) \):

\[
\tan(65^\circ) \approx 2.1445
\]

Now we can calculate \( d \):

\[
d = \frac{120}{2.1445} \approx 55.8 \text{ feet}
\]

Thus, the horizontal distance from the base of the tower to where the zip line ends is approximately **55.8 feet**.