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 Answer | Add Yours
(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.
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