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.

Question

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.

Answer 1

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.