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) 0 1 2 3 4 5 6 7 8 9
Height(m) 55 53 46 36 25 14 7 5 8 15
a. Find the amplitude, period, vertical translation, and phase shift for this function.
b. Determine the equation of the function in the form:
h(t) = asin [b(t – c)] + d.
c. What would the rest position of the pendulum be?
Can someone explain using degrees instead of pi please
Since your data do not show any symmetries or even the maximum or minimum, all we can do is pick four points and solve for a,b,c,d.
h(t) = a sin [b(t – c)] + d
Since the maximum height occurs at (0,55), I'd like to start with a cosine curve.
You don't actually say that the minimum height is h=5, but if we assume that, then we have
amplitude = (55-5)/2 = 25
center line at (55+5)/2 = 30
So, we can start with
h(t) = 25 cos(b(t-c)) + 30
See what you can do about the period and phase shift, yeah?
Come back if you get stuck, or my assumptions are wrong.
a. To find the amplitude, period, vertical translation, and phase shift for this function, we need to analyze the given data:
The amplitude is the maximum value the function reaches from its midline. Looking at the given heights, the maximum value reached is 55 meters, so the amplitude is 55.
The period is the time it takes for the motion to repeat. In this case, the motion seems to repeat every 8 seconds. So the period is 8.
The vertical translation is the vertical shift of the entire graph. Looking at the given heights, it seems to be shifted downward. The minimum value reached is 5 meters. Since the midline is halfway between the maximum and minimum values, the vertical translation is (55 + 5) / 2 = 30.
The phase shift is related to the horizontal shift of the graph. Looking at the given times, the maximum occurs at t = 0. So there is no phase shift, and c = 0.
b. Now, we can write the equation of the function in the form h(t) = asin[b(t – c)] + d:
Using the values we found:
amplitude = 55,
period = 8,
phase shift = 0,
vertical translation = 30.
The equation becomes:
h(t) = 55 * sin[(2π / 8)(t - 0)] + 30.
Simplifying further, we have:
h(t) = 55 * sin[(π / 4)t] + 30.
c. The rest position of the pendulum is the vertical position where the pendulum is at equilibrium. In this case, it is the vertical translation, which is 30 meters.
a. To find the amplitude, period, vertical translation, and phase shift of the function, we need to analyze the given data.
Amplitude: The amplitude represents half the distance between the maximum and minimum values of the function. Looking at the height values, we can see that the maximum value is 55 meters and the minimum value is 5 meters. Since the amplitude is half the difference between these values, we have an amplitude of (55-5)/2 = 25 meters.
Vertical translation: The vertical translation represents the average height or the midline of the function. To find this, we can take the average of the maximum and minimum values. (55 + 5)/2 = 30 meters.
Period: The period is the time it takes for one complete cycle of the function. Looking at the given data, we can see that the height values repeat after 9 seconds. Therefore, the period is 9 seconds.
Phase shift: The phase shift represents any horizontal translation of the function. From the data, the height starts at time t = 0. Therefore, there is no phase shift as the function starts at its original position.
b. The equation of the function in the form h(t) = asin[b(t – c)] + d, where a represents the amplitude, b represents the period, c represents the phase shift, and d represents the vertical translation.
Using the information we found earlier:
Amplitude = 25 meters,
Period = 9 seconds,
Phase shift = 0 seconds,
Vertical translation = 30 meters.
The equation of the function is:
h(t) = 25sin[(2π/9)(t - 0)] + 30.
c. The rest position of the pendulum can be determined by finding the average of the maximum and minimum heights. From the given data, the maximum height is 55 meters and the minimum height is 5 meters. Therefore, the rest position is the average of these values: (55 + 5)/2 = 30 meters.