Question

PLEASE HELP HURRY

The camp director is ready to purchase the cable. Use the distance between the trees and the change in height of the cable to determine the length of cable needed. Be sure to include:
• the required 5% slack in the line, and
• 6 extra feet of cable at each end to wrap around each tree.
Enter the total length, in feet, of cable needed.

slope constraint: the slope of the zip line should be 6 to 8 feet of vertical change for every 100 feet of horizontal change

Answer 1

Explanation:

To determine the length of cable needed for the zip line, we first need to calculate the horizontal distance between the two trees. Let's assume that the distance between the trees is d feet. We also need to calculate the vertical change in height of the cable between the two trees. Let's assume that the height difference is h feet.

Given the slope constraint, we know that for every 6 to 8 feet of vertical change, we need 100 feet of horizontal change. This means that the slope of the zip line should be between 6/100 and 8/100. We can use this information to set up a proportion:

vertical change / horizontal change = slope

h / d = 6/100 or h / d = 8/100

Solving for d in each of these equations, we get:

d = h / (6/100) = 100h / 6 = 50h / 3 (for the minimum slope of 6/100)

d = h / (8/100) = 100h / 8 = 25h / 2 (for the maximum slope of 8/100)

To account for the required 5% slack in the line, we need to increase the length of the cable by 5%. This means that the actual length of cable needed is:

L = 1.05d + 12

where 12 feet represents the 6 extra feet of cable needed at each end to wrap around each tree.

Substituting the expressions for d that we derived earlier, we get:

L = 1.05(50h/3) + 12 = (175h/6) + 12 (for the minimum slope of 6/100)

L = 1.05(25h/2) + 12 = (131h/40) + 12 (for the maximum slope of 8/100)

Therefore, the total length of cable needed for the zip line depends on the height difference between the trees and the desired slope of the line. For example, if the height difference is h = 50 feet and we want to use the minimum slope of 6/100, then the total length of cable needed would be:

L = (175(50)/6) + 12 = 1462.5 feet

If we want to use the maximum slope of 8/100, then the total length of cable needed would be:

L = (131(50)/40) + 12 = 73.2 + 12 = 85.2 feet

Note that these values are approximations and do not take into account factors such as the weight of the rider or the tension of the cable.

Answer 2

To calculate the length of cable needed, we first need to determine the length of the zip line itself. We can use the distance between the trees and the change in height of the cable to calculate the horizontal and vertical distances.

Let's say the distance between the trees is 200 feet and the change in height is 80 feet. To ensure a 6 to 8 feet of vertical change for every 100 feet of horizontal change, we need to limit the horizontal distance to between 750 and 1000 feet.

We can use the Pythagorean theorem to calculate the length of the zip line:

horizontal distance = √(100^2 - 80^2) = 60 feet

total length of zip line = √(200^2 + 60^2) = 208.44 feet

To account for the required 5% slack in the line, we need to add 5% to the total length:

total length with slack = 208.44 + (0.05 * 208.44) = 218.86 feet

To wrap around each tree, we need an additional 6 feet at each end, so we need to add 12 feet to the total length:

total length with slack and extra cable = 218.86 + 12 = 230.86 feet

Therefore, we need a total length of 230.86 feet of cable.