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 16 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).
Enter the exact answers.
Amplitude: A=
meters
Midline: h=
meters
Period: P=
minutes
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).
Hints:
What is the value of h(0)?
Is this the maximum value of h(t), the minimum value of h(t), or a value between the two?
The function sin(t) has a value between its maximum and minimum at t=0 , so can h(t) be a straight sine function?
The function cos(t) has its maximum at t=0, so can h(t) be a straight cosine function?
c. If the Ferris wheel continues to turn, how high off the ground is a person after 60 minutes?
check the related questions below. This has come up several times now.
a. To find the amplitude, midline, and period of h(t), we need to analyze the given information.
The diameter of the Ferris wheel is 28 meters, which means the radius is half of that, 28/2 = 14 meters. Since the maximum height of the Ferris wheel is the radius plus the height of the platform, the amplitude A is 14 + 1 = 15 meters.
The midline h is the average of the maximum and minimum heights, which is equal to the height of the platform, 1 meter.
The period P is the time it takes for one full revolution, which is given as 16 minutes.
Therefore:
Amplitude: A = 15 meters
Midline: h = 1 meter
Period: P = 16 minutes
b. Since the Ferris wheel starts spinning at time t = 0 and a person has just boarded from the platform, we can assume that at t = 0, the person's height is at the minimum value, which is 1 meter.
The height function h(t) can be modeled using a cosine function because at t = 0, the cosine function has its maximum value.
So, the formula for the height function h(t) is:
h(t) = A * cos((2π/P) * t) + h
Substituting the values we found earlier:
h(t) = 15 * cos((2π/16) * t) + 1
c. To find the height of a person after 60 minutes, we substitute t = 60 into the height function:
h(60) = 15 * cos((2π/16) * 60) + 1
Now we can calculate the height using a calculator:
h(60) ≈ 15 * cos(7.5π) + 1
h(60) ≈ 15 * (-1) + 1
h(60) ≈ -14 + 1
h(60) ≈ -13 meters
Therefore, a person is approximately 13 meters below the ground after 60 minutes.
a. To find the amplitude, midline, and period of h(t), we need to analyze the properties of the given information.
Amplitude (A) represents half the distance between the maximum and minimum values of the function. In this case, the height of the Ferris wheel is measured from the ground, so the maximum height would be when the person is at the topmost point of the Ferris wheel. Since the Ferris wheel has a radius of half the diameter (14 meters), the maximum height above the ground is 14 + 1 = 15 meters. Therefore, the amplitude is A = 15 meters.
Midline (h) represents the average height of the function. Since the person boards the Ferris wheel from a platform that is 1 meter above the ground, the midline would be at a height of 1 meter. Therefore, the midline is h = 1 meter.
Period (P) represents the length of one complete cycle of the function. In this case, the Ferris wheel completes one full revolution in 16 minutes. Since a full revolution corresponds to a complete cycle of the function, the period is P = 16 minutes.
Therefore, the answers are:
Amplitude: A = 15 meters
Midline: h = 1 meter
Period: P = 16 minutes
b. To find a formula for the height function h(t), we can utilize a sine function, as it has a midline at the height of the person when they board the Ferris wheel.
The generic equation for a sine function is h(t) = A sin(B(t - C)) + D, where:
A represents the amplitude
B represents the period
C represents the horizontal shift
D represents the vertical shift
From part a, we know that the amplitude is 15 meters (A = 15) and the period is 16 minutes (P = 16). We also know that when t = 0 (the start of the ride), the height is 1 meter above the ground. This gives us a vertical shift of 1 meter (D = 1).
Given this information, the formula for the height function becomes:
h(t) = 15 sin((2π/16)(t - C)) + 1
To find the horizontal shift, we need to identify when the height function reaches its maximum value. In this case, the height is maximum at the six o'clock position, which corresponds to t = 0. Therefore, t = 0 would be the maximum point of the sine function. This indicates that the horizontal shift (C) is 0.
Therefore, the formula for the height function h(t) is:
h(t) = 15 sin((π/8)t) + 1
c. To find how high off the ground a person is after 60 minutes, we simply need to substitute t = 60 into our height function:
h(60) = 15 sin((π/8)(60)) + 1
Calculating this expression will give us the height of the person after 60 minutes.