Question

You want to set up a zipline between a parking lot and the top of a cliff that is the end of a popular vista point hike. You want to charge people to take the zipline back to the parking so that they do not have to hike all the way. You have figured out that the distance to the base of the cliff is 2800 feet and the angle of elevation from the parking lot to the vista point is 30 degrees. How long does the cable have to be to make the zipline?

a) 2000 feet
b) 3233.16 feet
c) 1400 feet
d) 5600 feet

Answer 1

To calculate the length of the zipline cable, use trigonometric functions with the given distance and angle of elevation. The correct length is approximately 3233.16 feet, which is option (b).

Step-by-step explanation:

The student is trying to determine the length of a cable for a zipline using trigonometry. Knowing the zipline is 2800 feet from the base of the cliff and that the angle of elevation is 30 degrees, the length of the zipline (hypotenuse of the triangle) can be calculated using the cosine function.

1. Set up the equation using the cosine formula: cos(θ) = adjacent side / hypotenuse.
2. Plug in the known values: cos(30°) = 2800 / hypotenuse.
3. The cosine of 30 degrees is √3/2, so the equation is √3/2 = 2800 / hypotenuse.
4. Rearrange the equation to solve for the hypotenuse: hypotenuse = 2800 / (√3/2).
5. Calculate the hypotenuse: hypotenuse ≈ 3233.16 feet.

Therefore, the correct length of the cable for the zipline is approximately 3233.16 feet, which corresponds to option (b).