Math Grade 11, Applications
- At Canada's Wonderland, a thrill seeker can ride the Xtreme
Skyflyer. This is essentially a large pendulum of which the rider is the bob. The height of the rider is given for various times:
Time(s)0123456789Height(m) 55 53 46 36 25 14 7 5 8 15
- Create a graph of the height of the pendulum with respect to time.
- Find the amplitude, period, vertical translation, and phase shift for this function. [Note: the table does not follow the bob through one complete cycle, so some thought will be required to answer this question.]
- Determine the equation of the function in the form:
h(t) = asin[b(t - c)] + d.
- How could the amplitude be determined without creating the graph or finding the function?
- What would the rest position of the pendulum be?
- What is the maximum displacement for this pendulum?
- The time for one complete cycle is the period. How long
would it take to complete 15 cycles?
(1) You will need to plot the points
(2) Assuming that the highest value given is the maximum and the lowest value given is the minimum:
(a) The amplitude is `A=("highest"-"lowest")/2=(55-5)/2=25` This is the maximal distance of a point from the midline.
(b) The vertical translation is 30. The midline for the sine function is y=0. Here the function goes from y=5 to y=55, so the midline is `y=(55+5)/2=30` (the average of the extreme values.)
(c) The graph appears to be a cosine graph. To write as a sine function note that we must shift one fourth of the period to the left.
The period is the time to complete one full cycle -- it takes 7 seconds to get from highest to lowest, so the period is 14 seconds.
The phase shift (horizontal translation of the graph) is `14/4=3.5`
(3) h(t)=asin[b(t-c)]+d -- a is the amplitude; the period is found by `p=(2pi)/b` so `b=(2pi)/p=(2pi)/14=pi/7` ; c is the horizontal translation (phase shift) so c=-3.5 (moving to the left); and d is the vertical translation so d=30.
The function is `h(t)=25sin[pi/7(t+3.5)]+30`
(4) The amplitude is `a=("highest"-"lowest")/2`
(5) The rest position of the pendulum would be the lowest height possible so h=5m
(6) The maximum displacement is 55m-5m=50m
(7) The period is 14sec. 15 cycles takes 210 seconds or 3.5 minutes.
** My calculator gives `y~~25.072sin(.459x+1.52)+29.949` using a regression for sine.