A Ferris wheel has a radius of 10 meters and is revolving 6 times each minute (wheel's frequency.) The wheel's center is 12 meters from the ground. If time starts on the ground, then find a sine function that shows the height from the ground for a car on the Ferris wheel at any time.
6 rotations take 1 minutes
1 rotation takes 1/6 minutes
2π/k = 1/6
k = 12π
let's start using only amplitude and frequency to get
height = 10 sin ( 12πt)
We want to move the curve up 12 units
height = 10 sin (12πt) + 12
so now we have to have a horizontal shift so that when t = 0, the height will be 2
2 = 10 sin 12π(0 + d) + 12
-1 = sin 12πd
now we know that sin 3π/2 = -1
then 12πd = 3π/2
d = 1/8
so one version of our graph would be
height = 10 sin 12π(t + 1/8) + 12
testing: if t = 1/12, we should be at our highest point of 22
if t = 1/24
height = 10 sin 12π(1/12+1/8) + 12
=10 sin 12π(5/24) + 12
= 10 sin 5π/2 + 12
= 10(1) + 12
= 22
confirmation here:
http://www.wolframalpha.com/input/?i=y+%3D+10+sin+%2812π%28t+%2B+1%2F8%29%29+%2B+12++for+0+%3C+t+%3C+.166666
To find a sine function that represents the height of a car on the Ferris wheel at any time, we can use the formula:
h(t) = A * sin(ωt + φ) + k
Where:
- h(t) is the height of the car at time t
- A is the amplitude of the function
- ω is the angular frequency of the function
- φ is the phase shift of the function
- k is the vertical shift of the function
Now, let's analyze the given information to determine the values of A, ω, φ, and k for our sine function.
1. Amplitude (A):
The amplitude of a sine function represents half the distance between the maximum and minimum values of the function. In this case, the Ferris wheel has a radius of 10 meters, so the distance between the highest and lowest point is twice the radius, which is 20 meters. Therefore, the amplitude is A = 20/2 = 10 meters.
2. Angular Frequency (ω):
The angular frequency of a sine function represents how quickly the function oscillates or completes a full cycle. In this case, the Ferris wheel completes 6 revolutions per minute, which means it completes 6 cycles in 1 minute. Therefore, the angular frequency is ω = 2π * 6 = 12π rad/min.
3. Phase Shift (φ):
The phase shift of a sine function represents the horizontal shift of the function from its standard position. In this case, time starts on the ground, so the initial position of the car is when t = 0. At this point, the car is at its maximum height, so the phase shift is φ = 0.
4. Vertical Shift (k):
The vertical shift of a sine function represents the vertical displacement of the function from its standard position. In this case, the center of the Ferris wheel is 12 meters from the ground, so the vertical shift is k = 12 meters.
Now we have determined all the required values. Substituting these values into the sine function formula, we get:
h(t) = 10 * sin(12πt) + 12
Therefore, the sine function that represents the height from the ground for a car on the Ferris wheel at any time is h(t) = 10 * sin(12πt) + 12.