The diameter of the wheel is 165 feet, it rotates at 1.5 revolutions per minute, and the bottom of the wheel is 9 feet 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.
To solve this problem, we need to consider the height at any time t during the ride. Let's break it down step by step:
1. Firstly, let's find the equation to represent the passenger's angular position at any time t.
The wheel completes 1.5 revolutions (or 1.5 * 360 degrees) in one minute. Since the passenger starts at the bottom, we can assign the initial angular position as 0 degrees.
The angular position (θ) of the passenger at any time t can be calculated using the formula:
θ = ωt
Where:
θ - angular position (in degrees)
ω - angular velocity (in degrees per minute)
t - time (in minutes)
Since the wheel completes 1.5 revolutions per minute, the angular velocity (ω) is:
ω = 1.5 * 360 = 540 degrees per minute
Therefore, the equation for the angular position is:
θ = 540t
2. Next, we need to determine the height at any given angular position θ.
Since the diameter of the wheel is 165 feet, the radius (r) can be calculated as half the diameter, which is:
r = 165 / 2 = 82.5 feet
The height above the ground (h) can be calculated using the formula:
h = r - rsin(θ)
Where:
h - height above the ground
r - radius of the wheel
θ - angular position (in radians)
To calculate the height, we need to convert the angular position from degrees to radians:
θ_rad = θ * π / 180
Now we can substitute the values in the formula:
h = 82.5 - 82.5sin(θ_rad)
h = 82.5 - 82.5sin(θ * π / 180)
Therefore, the equation that gives the passenger's height above the ground at any time t is:
h = 82.5 - 82.5sin(540t * π / 180)