Question

A zip line that is 170 ft long runs off the edge of a cliff and makes an angle of 29° with the ground below. How far away from the wall of the cliff is the cable attached to the ground? Round to the nearest hundredths (2 decimal places).(2 points) The cable is feet away from the base of the cliff.

Answer 1

To find out how far away from the base of the cliff the zip line (cable) is attached to the ground, we can use trigonometry. Specifically, we can use the cosine function, which relates the adjacent side and hypotenuse of a right triangle.

In this situation:
- The length of the zip line (hypotenuse) is 170 ft.
- The angle with the ground is 29°.
- We want to find the horizontal distance from the base of the cliff to the point where the zip line is anchored on the ground (the adjacent side).

Using the cosine function:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]

Plugging in the values:
\[
\cos(29°) = \frac{x}{170}
\]
where \( x \) is the distance from the wall of the cliff to the attachment point on the ground.

Now, let's solve for \( x \):
\[
x = 170 \cdot \cos(29°)
\]

Calculating \( \cos(29°) \):
\[
\cos(29°) \approx 0.8746
\]

Now, substitute this value back into our equation:
\[
x \approx 170 \cdot 0.8746 \approx 148.66
\]

Rounding to the nearest hundredths, the distance from the wall of the cliff is:
\[
\boxed{148.66 \text{ ft}}
\]