Expert: Alon Mandes - 10/12/2009

Question
Here's some background information for the problem:
 One day w decide to go to six flags. They decide to do the bungee jump. They start at a height of 80 feet. Once they have fallen, they gently swing back and forth, up and down (like a pendulum). The closest they get to the ground is 10 feet, they are furthest away from the ground at 25 feet, and it takes them 2 seconds to get from "up" to "down" and back to "up" again.

 A) make a table that compares the riders' height(from the ground) over time.
   How would u go about doing this?
   B) Write an equation for the function expressing the riders' height from the ground (h) as a function of time (t).
   I assume u have to find the amplitude, the period, and the mid line, using the equation: y= a sin(bx)+d
but i don't know how to do this with the given information.
   *** My paper is due tomorrow so i would greatly appreciate it if it u could get it to me today. I am having a lot of trouble with this and i would just like to thank you so much ahead of time for taking the time to help me with this problem. Thank you soo much!

Answer
Hello Nicolette,
I will solve section B. As for section A, you can see it more clearly through the formula.
Yes, the height will be expressed in terms of periodic function, such as sin %26 cos . The
appropriate one is cos %26 not sin. The reason for that, is due to the initial condition :
We demand that in t=0 , h=80 . Since we know that sin(0)=0 %26 cos(0)=1 , the cosine function
will do the work. So, we assume that h(t)=aCos(bt)+d . 1st we must find a , then d %26 then b.
Finding a :
---------
The function cos(t) varies from -1 to 1 gaining a total amplitude of 2 . While the function
35Cos(t) varies from -35 to 35 gaining an amplitude of 70 . Since the middle of the limits
of the heights of the swinging is  : (80-10)/2 , therefore 35Cos(bt) is our 1st step.
Finding d :
---------
When t=0, the height must be 80. So, 35Cos(0)+d must be equal 80 . This means 35+d=80 -->
d=45. Therefore, our equation will become h(t)=35Cos(bt)+45 . Let's chick if , the minimum of
this function reach the height of 10  : The minimum will be achieved when Cos(bt)=-1, giving
us : -35+45 =10 Yes, perfect !
Finding b :
---------
To find b, we use the periodic information. The period time is 2 seconds. That means in t=2
h(2) must be equal to the original height , which is 80 . Let's write this condition in
mathematical way : 35Cos(2b)+45=80 --> 35Cos(2b)=35 --> Cos(2b)=1 . Now we ask ourselves
when does Cos function get value of 1 ? the answer is , when the argument inside the cosine
is 2π or 360 degrees . Thus, 2b=2π --> b= π .

Now we can write analytically the equation of height :
h(t)=35Cos(πt)+45 .

Alon.