3. A Ferris wheel has a diameter of 16𝑚. At 𝑡 = 0 a passenger is at the lowest point on the wheel. At time 𝑡 = 40𝑠, a passenger is at the top of the wheel, 18 𝑚 above the ground.

Question

a. provide points to Sketch one cycle on the graph.
b. State the equation of the graph.
c. At what times during in the first 4 minutes can the passenger get off the wheel safely if it stopped?
d. Could the passenger get off of the wheel safely at 5 min if it stopped then? Explain.

Answer 1

a. Points to sketch one cycle on the graph:
1. Lowest point: (0, -16)
2. Highest point: (40, 2)
3. Midpoint on the left side: (20, -8)
4. Midpoint on the right side: (60, -8)

b. Equation of the graph:
The equation of the Ferris wheel can be modeled by a cosine function. The general form of a cosine function is:
y = A cos(B(x - C)) + D
Where:
A = amplitude
B = 2π/period
C = horizontal shift
D = vertical shift

In this case, the amplitude is 8 (half of the diameter), the period is 80 seconds (time to complete one cycle), the horizontal shift is 0, and the vertical shift is 0. Therefore, the equation of the Ferris wheel is:
y = 8 cos((2π/80)x)

c. At what times during the first 4 minutes can the passenger get off the wheel safely if it stopped?
The safe times for the passenger to get off the wheel are when the passenger is at the lowest point or the highest point. In the first 4 minutes (240 seconds), the passenger can safely get off the wheel at times 0 seconds, 40 seconds, and 160 seconds.

d. Could the passenger get off the wheel safely at 5 minutes if it stopped then? Explain.
At 5 minutes (300 seconds), the passenger would be at a height of 8 meters, which is not the lowest or highest point. Therefore, it is not safe for the passenger to get off the wheel at 5 minutes. The safe times are when the passenger is at the lowest or highest point.