A Ferris wheel is 28 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 8 minutes. The function h(t) gives a person’s height in meters above the ground t minutes after the wheel begins to turn.
a. Find the amplitude, midline, and period of h(t) .
b.Assume that a person has just boarded the Ferris wheel from the platform and that the Ferris wheel starts spinning at time t=0. Find a formula for the height function h(t).
c. If the Ferris wheel continues to turn, how high off the ground is a person after 30 minutes?
a. The amplitude is the radius of the Ferris wheel, which is half the diameter. So, the amplitude is 28/2 = 14 meters. The midline is the height of the loading platform, which is 1 meter above the ground. The period is the time it takes for the Ferris wheel to complete one full revolution, which is 8 minutes.
b. Since the Ferris wheel completes one full revolution in 8 minutes, it means it goes from its lowest point to its highest point and back in that time. So, the height function h(t) will follow a sine curve. Given that the midline is 1 meter, the formula for the height function is:
h(t) = 14*sin((2π/8)t) + 1
c. To find the height after 30 minutes, we substitute t = 30 into the height function:
h(30) = 14*sin((2π/8)*30) + 1
Calculating this, we get:
h(30) ≈ 14*sin(2π*3.75) + 1
≈ 14*sin(7.5π) + 1
Now, since sin(7.5π) = 0, the height after 30 minutes will be equal to the midline:
h(30) ≈ 1 meter
Therefore, after 30 minutes, the person will be 1 meter above the ground.
a. To find the amplitude, midline, and period of h(t), we need to study the properties of a trigonometric function.
1. Amplitude:
The amplitude represents the maximum vertical distance from the midline. In this case, since the Ferris wheel has a diameter of 28 meters, the radius is half of that, which is 14 meters. Therefore, the amplitude is 14 meters.
2. Midline:
The midline represents the average height of the function. In this case, since the platform is 1 meter above the ground, the midline is 1 meter.
3. Period:
The period represents the time it takes for one complete cycle of the function. In this case, the Ferris wheel completes one full revolution in 8 minutes. Since the height function goes through one full cycle in the same amount of time, the period is 8 minutes.
b. We need to create an equation for the height function h(t) based on the information given.
Since the Ferris wheel starts at the six o'clock position, which is level with the loading platform, the initial height is 1 meter.
The height function can be represented using the cosine function:
h(t) = amplitude * cosine((2*pi / period) * t) + midline
Plugging in the values we found earlier:
h(t) = 14 * cosine((2*pi / 8) * t) + 1
c. To find a person's height after 30 minutes, we substitute t = 30 into the height function:
h(30) = 14 * cosine((2*pi / 8) * 30) + 1
Calculating this expression will give us the height off the ground after 30 minutes.
a. To find the amplitude, midline, and period of h(t), we can use the given information.
The amplitude of a periodic function is half the difference between the maximum and minimum values. Since the Ferris wheel has a diameter of 28 meters, the radius (and thus the amplitude) is 28/2 = 14 meters.
The midline of the function is the average value of the maximum and minimum values. In this case, the midline is the height of the platform, which is 1 meter above the ground.
The period of a periodic function is the length of one complete cycle. In this case, a cycle is completed when the Ferris wheel makes one full revolution, which takes 8 minutes. Therefore, the period is 8 minutes.
Therefore, the amplitude is 14 meters, the midline is 1 meter, and the period is 8 minutes.
b. To find a formula for the height function h(t), we need to consider that the starting position of the Ferris wheel corresponds to t=0, and that the six o'clock position (level with the loading platform) is the midline.
Considering the midline is 1 meter, we can use the amplitude to find the maximum height above the midline. Since the amplitude is 14 meters, the maximum height is 1 + 14 = 15 meters.
A cosine function is suitable for modeling the height of the Ferris wheel because it oscillates between the maximum and minimum values.
A general formula for a cosine function is: h(t) = A * cos(2π / T * (t - d)) + m
Where:
- A is the amplitude,
- T is the period,
- d is the horizontal shift (phase shift), and
- m is the midline.
In this case, A = 14, T = 8, d = 0 (since the Ferris wheel starts spinning at t=0), and m = 1.
Therefore, the formula for the height function h(t) is: h(t) = 14 * cos(2π / 8 * t) + 1.
c. To find how high off the ground a person is after 30 minutes, we can substitute t=30 into the height function h(t).
h(30) = 14 * cos(2π / 8 * 30) + 1 = 14 * cos(2π / 240 * 30) + 1.
Evaluating this expression would give you the height of the person after 30 minutes.
28 meters in diameter means the radius is 14 -- that's he amplitude
boarded from a platform that is 1 meter above the ground means the axle is 1+14 = 15 feet up
six o’clock position on the Ferris wheel is level with the loading platform means that at t=0 you have a minimum, so that's something like y = a-bcos(kt)
1 full revolution in 8 minutes means that the period is 8. Since cos(kt) has a period of 2π/k, that means that k = π/4
Now put that all together