1. Fun Village has North America’s largest Ferris wheel. The Ferris wheel has a diameter of 56 m, and one revolution took 2.5 min to complete. Riders would start at the bottom, which is 0.5 m above the ground, and could see Niagara Falls if they were higher than 50 m above the ground.
a) Create an equation that represents the height of a rider above the ground, as a function of time.
b) Using the equation above, determine the time(s) at which the rider reaches a height of 5 m during the second revolution.
a) Let's assume that the Ferris wheel starts at the bottom and completes one full revolution in 2.5 minutes. We can calculate the height of a rider above the ground at any given time using the equation for the height of a point on a circle:
h(t) = r * sin(θ)
In this equation, r is the radius of the Ferris wheel (half of the diameter, so 28 m) and θ is the angle measured in radians. In this case, we need to convert the time t to an angle θ.
To find the angle θ, we can use the formula for the angular velocity:
ω = 2π / T
Where ω is the angular velocity (in radians per minute) and T is the time period for one revolution (in minutes). In this case, ω = (2π) / 2.5.
Now we can substitute the values into the equation for the height:
h(t) = 28 * sin((2π / 2.5) * t)
b) To determine the time(s) at which the rider reaches a height of 5 m during the second revolution, we need to set h(t) equal to 5 and solve for t. Since one revolution takes 2.5 minutes, the second revolution starts at t = 2.5 minutes.
So we have:
5 = 28 * sin((2π / 2.5) * t)
To solve this equation, we can use inverse sine function:
sin^(-1)(5 / 28) = (2π / 2.5) * t
Now we can calculate t:
t = sin^(-1)(5 / 28) * (2.5 / (2π))
Using a calculator, we can find the value of t.
a) To create an equation that represents the height of a rider above the ground as a function of time, we can start by determining the period of the Ferris wheel's revolution.
The period is the time it takes for one complete revolution. Given that one revolution takes 2.5 minutes to complete, we can calculate the period (T) as:
T = 2.5 minutes
Now, let's define the equation for the height of the rider above the ground (h) as a function of time (t).
We know that the Ferris wheel has a diameter of 56 m, which means the radius (r) is half of that:
r = 56 m / 2 = 28 m
The height of the rider, h, can be calculated using the equation:
h = r * sin(2πt / T) + h0
Where:
- t represents the time elapsed since starting the ride
- π is a mathematical constant (approximately 3.14159)
- T is the period of one revolution (2.5 minutes or whatever units are used for time)
- r is the radius of the Ferris wheel
- h0 is the initial height of the rider above the ground (0.5 m, given in the question)
Therefore, the equation that represents the height of the rider above the ground as a function of time is:
h = 28 * sin(2πt / 2.5) + 0.5
b) To determine the time(s) at which the rider reaches a height of 5 m during the second revolution, we need to solve the equation for h = 5 and find the corresponding values of t.
Setting h = 5 in the equation:
5 = 28 * sin(2πt / 2.5) + 0.5
Now, we can solve this equation for t by isolating the variable:
28 * sin(2πt / 2.5) = 4.5
sin(2πt / 2.5) = 4.5 / 28
Using an inverse sine function, we can determine the angle whose sine is equal to 4.5 / 28:
2πt / 2.5 = arcsin(4.5 / 28)
Next, solve for t by multiplying both sides by 2.5 and dividing by 2π:
t = (2.5 * arcsin(4.5 / 28)) / (2π)
This will give you the value(s) of t when the rider reaches a height of 5 m during the second revolution.