In 1893, the city of Chicago put on a World's Fair to celebrate the 400th anniversary of Columbus' trip to the Americas. The most spectacular at this fair was a huge Ferris Week built by an engineer named George Ferris. This remarkable structure had a diameter of 250 ft, was 30 ft wide and weighed about 4000 tons. There were 36 enclosed cars on the ride that seated a total of 1440 people, although as many as 2000 people could be accommodated through standing room. When a person was at the top of the ferris wheel, they were higher than a 20 story building.

Suppose that the diameter of this wheel is 250 ft and that the bottom of the wheel is 10 ft off the ground to assure that the cars do not strike the ground. Suppose also that when the car is in continuous motion it takes 60 seconds for it to make a complete revolution. As you watch a particular car travel around the wheel, you start a stopwatch and observe that the car reaches its highest point after 10 secongs.

a. draw a graph of the height of the car vs. the time since the stopwatch started

b. find an equation that models the height of the car above the ground versus time since you starter the stopwatch

c. how high is the car when the watch started? after 30 seconds? one minute? 45 seconds? 53 seconds?

d. what are the next two times that the car will be at its highest point?

e. when is the first time the car is 200 feet from the ground? what is the next time? the third time?

a. To draw a graph of the height of the car vs. time, we will use the information given. We know that the car takes 60 seconds to make a complete revolution, so we can divide the 60 seconds into four equal parts, each representing a quarter of the revolution. Since it takes 10 seconds for the car to reach its highest point, the graph will start at 0 seconds with the car at ground level, then rise to its highest point at 10 seconds, and then subsequently fall back to ground level at 20 seconds. The graph will then repeat this pattern for the remaining two quarters of the revolution.

b. To find an equation that models the height of the car above the ground vs. time, we can use a sine or cosine function since these functions have a period of 60 seconds. We can represent the height of the car as a function of time using the equation:

h(t) = A * sin(B * (t + C)) + D

A represents the amplitude of the function, which in this case is half of the diameter of the wheel (250/2 = 125 ft).
B represents the frequency of the function, which is given by 2π divided by the period of 60 seconds. Therefore, B = 2π/60 = π/30.
C represents the phase shift of the function, which will be 0 in this case since we are measuring time since the stopwatch started.
D represents the vertical shift of the function, which is the initial height of the car above the ground, given as 10 ft.

Therefore, the equation that models the height of the car above the ground vs. time since the stopwatch started is:

h(t) = 125 * sin(π/30 * t) + 10

c. To find the height of the car at different times, we can substitute the values of t into the equation we derived in part b.

- When the watch started (t = 0 seconds): h(0) = 125 * sin(π/30 * 0) + 10 = 125 * sin(0) + 10 = 10 ft
- After 30 seconds (t = 30 seconds): h(30) = 125 * sin(π/30 * 30) + 10 = 125 * sin(π) + 10 = 125 * 0 + 10 = 10 ft
- After one minute (t = 60 seconds): h(60) = 125 * sin(π/30 * 60) + 10 = 125 * sin(2π) + 10 = 125 * 0 + 10 = 10 ft
- After 45 seconds (t = 45 seconds): h(45) = 125 * sin(π/30 * 45) + 10 = 125 * sin(3π/2) + 10 = 125 * (-1) + 10 = -115 ft
- After 53 seconds (t = 53 seconds): h(53) = 125 * sin(π/30 * 53) + 10 = 125 * sin(53π/30) + 10 ≈ 141.506 ft

d. To find the next two times that the car will be at its highest point, we need to determine when the sine function reaches its maximum value. The maximum value of a sine function is 1, so we can solve for t in the equation:

sin(π/30 * t) = 1

Simplifying, we have:

π/30 * t = π/2

t = (π/2) * (30/π) = 15 seconds

Therefore, the car will reach its highest point at 15 seconds, and the next time will be another 15 seconds later, at 30 seconds.

e. To find the time when the car is 200 feet from the ground, we need to solve the equation:

125 * sin(π/30 * t) + 10 = 200

Simplifying, we have:

125 * sin(π/30 * t) = 190

sin(π/30 * t) ≈ 1.52

Since the sine function oscillates between -1 and 1, there are no real solutions for t in this equation. Therefore, the car never reaches a height of 200 feet.

If you have any further questions, feel free to ask.