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) 1400 feet
b) 2000 feet
c) 2800 feet
d) 3233.16 feet

Answer 1

Using the cosine function for a 30-degree angle in a right triangle, the zipline cable length needed to connect the parking lot to the cliff top is approximately 3233.16 feet, corresponding to option d).

Step-by-step explanation:

To determine the length of the cable needed for a zipline from a parking lot to the top of a cliff, we can use trigonometric principles. Given the distance to the base of the cliff is 2800 feet and the angle of elevation is 30 degrees, we are dealing with a right triangle where the distance to the base represents the adjacent side to the angle of elevation, and the length of the cable will be the hypotenuse.

Using the cosine function for a 30-degree angle in a right triangle (cos(30) = adjacent/hypotenuse), we can solve for the hypotenuse (cable length) as follows:

cos(30 degrees) = 2800 feet / hypotenuse
hypotenuse = 2800 feet / cos(30 degrees)
hypotenuse ≈ 2800 feet / 0.866 = 3233.16 feet

Therefore, the cable has to be 3233.16 feet long, which corresponds to option d).