A popular extreme activity is bungee jumping. A person whose ankles are attached to a bungee cord, jumps off a ledge and bounces upside-down in mid-air. Suppose a person jumps off the ledge and falls 122 ft before rebounding. Consider the point at which the person rebounds to be t=0. The person hen rebounds to a distance of 46 feet from the ledge after 3 seconds (assume there is no resistance in the bungee cord).

1. Write a function that represents the person's vertical distance from the ledge as a function of time
2. Graph the function you wrote in question 1
3. If the bungee cord's resistance slowed the person's bounce so that they bounced 7.6 feet less each cycle, how long would it take for the person to stop bouncing?
4. How long is the bungee cord if the conditions in question 3 are true?
5. sketch a graph of the situation if the conditions in question 3 are true

requirements: answers for 1, 3, 4 must include all work.

1. To write a function representing the person's vertical distance from the ledge as a function of time, we need to consider two phases: the falling phase and the bouncing phase.

a) Falling phase: The person is falling for the first 122 ft before rebounding. Assuming there is no resistance in the bungee cord, the falling phase can be represented by a linear function (since the person falls at a constant rate):
f(t) = -16t^2 + 122

b) Bouncing phase: After rebounding, the person bounces back up to a distance of 46 ft from the ledge after 3 seconds. Assuming the bounces are equally spaced and each bounce is 7.6 ft shorter than the previous one, we can use a geometric sequence to represent the bouncing phase.
The common ratio (r) in our case is given by: r = (46 - 7.6) / 46 = 0.835
The term a is 46 ft.
The general formula for the bouncing distance as a function of time (t) is: g(t) = a * r^t

Combining both phases, the function representing the person's vertical distance from the ledge at any given time is:
h(t) = -16t^2 + 122 + 46 * 0.835^t

2. To graph the function h(t), plot the values of t on the x-axis and the corresponding values of h(t) on the y-axis. Choose a suitable range for t based on the given information (e.g., from t = 0 to t = 10 seconds) and calculate the corresponding values of h(t) using the function. Then, plot the points and connect them with a smooth curve.

3. In order to find when the person stops bouncing, we need to determine the value of t for which h(t) becomes zero. Set h(t) = 0 and solve for t:
-16t^2 + 122 + 46 * 0.835^t = 0

This equation cannot be solved algebraically. We need numerical methods or approximations to find the solution. One way is to use numerical techniques like the Newton-Raphson method or graphing and finding the intersection point where h(t) crosses the x-axis.

4. Once we find the time at which the bouncing stops, we can substitute this value into the function for the bouncing phase (g(t)) to find the length of the bungee cord. Since the time when bouncing stops represents the end of the last bounce, the length of the bungee cord is given by the sum of all the previous bounces:
Bungee Cord Length = a + a * r + a * r^2 + ... + a * r^(n-1)

Substitute the values of a, r, and n into this formula to find the length of the bungee cord.

5. To sketch a graph of the situation when the conditions in question 3 are true, you need to modify the graph of h(t) by reducing the height of each bounce by 7.6 ft while keeping the same bouncing pattern. To do this, subtract 7.6 ft from the values of a in the bouncing phase function (g(t)), and then plot the modified points and connect them with a smooth curve.