The diameter of the wheel is 165 ft, it rotates at 1.5 revolutions per minute, and the bottom of the wheel is 9 ft above the ground. Find an equation that gives a passenger's height above the ground at any time t during the ride. Assume the passenger starts the ride at the bottom of the wheel.
1.5 revs -- 60 sec
1 rev --- 40 seconds
So the period is 40 seconds
2π/k = 40
k = π/20
amplitude = 165/2 = 82.5
So let's start with
height = 82.5 sin (π/20)t, where t is in seconds
But that would give us a min of 82.5, and we want to have a min of 9
so height = 82.5 sin (π/20)t + 91.5
But we know that for t = 0 , we should get 9
so we have to apply a phase shift
let height = 82.5 sin (π/20)(t + c) + 91.5
when t = 0, height = 9
9 = 82.5 sin(π/20)(t+c) + 91.5
-1 = sin (π/20)(c)
we know sin 3π/2) = -1
so (π/20)(c) = 3π/2
c/20 = 3/2
c = 30
height = 82.5 sin (π/20)(t+30) + 91.5
checking:
at t = 0 we should get height = 9 , we do!
at t = 10 we should get 91.5 , we do!
at t= 20 , we should get a max of 174 , we do!
at t = 30, we should get 91.5 again, we do!
at t = 40 we should be back to 9, we do!
This is just one such equation, we could have used a cosine function in a similar method.
To find an equation that gives the passenger's height above the ground at any time during the ride, we can break the problem down into smaller steps:
Step 1: Determine the equation for the wheel's rotation.
Step 2: Convert the rotation of the wheel to an angle.
Step 3: Find the vertical distance traveled by the passenger.
Step 4: Determine the passenger's height above the ground at any time.
Let's tackle each step one by one:
Step 1: Determine the equation for the wheel's rotation.
The wheel makes 1.5 revolutions per minute. A revolution is equal to 360 degrees. Therefore, the wheel rotates at a constant rate of 1.5 * 360 = 540 degrees per minute.
Step 2: Convert the rotation of the wheel to an angle.
To convert the rotation of the wheel to an angle at any time t, we can use the equation: angle = (rotation rate) * t.
In this case, the rotation rate is 540 degrees per minute.
So, the equation for the angle at any time t is:
angle = 540t
Step 3: Find the vertical distance traveled by the passenger.
Since the diameter of the wheel is 165 ft, the radius is half of that, which is 82.5 ft.
As the wheel rotates, the passenger travels a circular path around the center of the wheel.
The vertical distance traveled by the passenger is equal to the radius multiplied by the sine of the angle.
So, the equation for the vertical distance traveled by the passenger is:
vertical distance = radius * sin(angle)
vertical distance = 82.5 * sin(540t)
Step 4: Determine the passenger's height above the ground at any time.
The passenger starts at the bottom of the wheel, where the height above the ground is 9 ft.
The passenger's height above the ground at any time t is equal to the sum of the starting height (9 ft) and the vertical distance traveled by the passenger.
So, the equation for the passenger's height at any time t is:
height = 9 + vertical distance
height = 9 + 82.5 * sin(540t)
Therefore, the equation that gives the passenger's height above the ground at any time t during the ride is:
height = 9 + 82.5 * sin(540t)