A carnival Ferris wheel with a radius of 7 m makes one complete revolution every 16s. The bottom of the wheel is 1.5 m above the ground.
Find the amplitude, vertical shift, phase shift and k value and period. Do all calculations and then make your final equation.
Predict how the graph and equation will change if the Ferris wheel turns more slowly.
Please stick to one name, don't switch names from one post to another.
this question is similar to the one you posted as Katlyn
since the radius is 7 m, a = 7
remember : period = 2π/k , where k is the coefficient of your trig function ... sin (kt)
so 2π/k = 16
16k = 2π
k = 2π/16 = π/8
so far we have height = 7 sin (π/8)t
the way the curve is, and ground is zero, the lowest point would be -7, but we want the lowest point to be 1.5, so wouldn't we have to raise the curve up 8.5 ?
so far we have : height = 7 sin (πt/8) + 8.5
since you did not state where you want the bottom of the wheel to be at a given time, I cannot find the phase shift
As the wheel slows down, the period would become longer, and the sine curve would be stretched horizontally, nothing happens vertically.
I have 2 names krystien and katlyn sometimes katlyien
To find the amplitude, we need to determine the maximum height reached by the Ferris wheel. Since the bottom of the wheel is 1.5 m above the ground, the maximum height will be the radius of the wheel plus the vertical shift. Therefore, the amplitude is 7 m + 1.5 m = 8.5 m.
The vertical shift is 1.5 m.
The period can be determined using the fact that the wheel takes 16 seconds to complete one revolution. Since one revolution is equivalent to 2π radians, the period is equal to the time taken for one complete cycle. Therefore, the period is 16 s.
The phase shift is 0, as there is no horizontal displacement of the wheel.
The equation describing the height of the Ferris wheel as a function of time is given by:
h(t) = A*sin(B(t-C))+ D
where:
A = amplitude, which is 8.5 m
B = 2π divided by the period, which is 2π/16 = 1/8
C = phase shift, which is 0
D = vertical shift, which is 1.5 m
Putting all the values together, the equation becomes:
h(t) = 8.5*sin((1/8)t) + 1.5
If the Ferris wheel turns more slowly, the period of the function would increase. Therefore, the value of B, which represents the frequency, would decrease. This would result in a longer time taken for the wheel to complete one cycle. Consequently, the graph would show a less steep wave with a longer wavelength. The function would also oscillate at a slower rate.
To find the amplitude, vertical shift, phase shift, and k value, we need to make some observations first.
1. Amplitude: The Ferris wheel is a perfect circle, and the radius represents the maximum distance from the center to the top or bottom. Hence, the amplitude is equal to the radius of the Ferris wheel, which is 7 meters.
2. Vertical shift: The bottom of the wheel is 1.5 meters above the ground. This means the graph will be shifted vertically upwards by 1.5 meters. Therefore, the vertical shift is +1.5.
3. Phase shift: The phase shift occurs when the graph starts at a different point on the x-axis. In this case, since the Ferris wheel starts at the bottom, the phase shift is 0. We don't need to account for any horizontal shift in the equation.
4. K value: The k value represents the frequency of oscillation. Since the Ferris wheel makes one complete revolution every 16 seconds, the period is 16 seconds. The k value, which is related to the period, can be calculated as 2π divided by the period. So, k = 2π/16 = π/8.
Now, let's summarize our findings and write the equation for the motion of the Ferris wheel:
Amplitude = 7
Vertical shift = +1.5
Phase shift = 0
k value = π/8
Period = 16
The equation will be of the form:
y = A sin(kx + φ) + c
Substituting the values:
y = 7 sin(π/8 x) + 1.5
Now, let's discuss how the graph and equation will change if the Ferris wheel turns more slowly.
If the Ferris wheel turns more slowly, the period will increase. This means the k value, which is inversely proportional to the period, will decrease. As k decreases, the oscillations will occur more slowly, causing the graph to stretch horizontally.
In terms of the equation, the k value in the trigonometric function will decrease, resulting in a slower oscillation. The amplitude, vertical shift, and phase shift will remain the same unless specifically mentioned.
Therefore, the equation will have a smaller k value, while the amplitude, vertical shift, and phase shift will remain unchanged. The graph will have a wider wave form compared to the original, indicating slower oscillations.