Ferris Wheel Heights and a Tour Guide's Claim
A Ferris wheel has a radius of 12 meters and its center is 14 meters above the ground. Riders board at the lowest point, where the height is 2 meters. The wheel rotates counterclockwise at a constant rate, completing one revolution every 40 seconds. A rider's height h(t), in meters above the ground t seconds after boarding, can be modeled with a cosine function. The operator wants to know when a rider is at least 20 meters above the ground during the first revolution, because that is when the best view is available. A tour guide claims that riders spend half of each revolution above 20 meters, since 20 meters is above the center height. The maintenance crew also asks how fast the rider's height is changing at t=10 seconds, but the operator only needs the times and the fraction of the ride for now.
1. Write the height function h(t) and explain how the amplitude, midline, and period come from the physical data.
Read the worked answer
The rider starts at the lowest point, so a negative cosine is appropriate: h(t) = midline - amplitude * cos(2pit/period). The midline is the center height, 14 meters. The amplitude is the radius, 12 meters. The period is 40 seconds. Thus
h(t) = 14 - 12 cos(2pit/40) = 14 - 12 cos(pi*t/20).
At t=0, h=14-12=2 meters, matching boarding. At t=20, h=14+12=26 meters, the top. At t=40, the rider returns to 2 meters.
2. Solve h(t) >= 20 for the first revolution and give the time interval.
Read the worked answer
Set 14 - 12 cos(pit/20) >= 20. Then -12 cos(pit/20) >= 6, so cos(pit/20) <= -1/2. On one revolution, the angle theta = pit/20 runs from 0 to 2pi. Cosine is at most -1/2 on [2pi/3, 4*pi/3]. Therefore
2pi/3 <= pit/20 <= 4*pi/3.
Dividing by pi and multiplying by 20 gives 40/3 <= t <= 80/3, or approximately 13.3 <= t <= 26.7 seconds. The rider is at least 20 meters above the ground during that interval.
3. Evaluate the tour guide's claim that riders spend half of each revolution above 20 meters.
Read the worked answer
The interval above 20 meters lasts from 40/3 to 80/3 seconds, a duration of 40/3 seconds, about 13.3 seconds. One revolution lasts 40 seconds, so the fraction is (40/3)/40 = 1/3, not 1/2. The guide's reasoning is flawed because the center height is 14 meters, not 20 meters. The condition h >= 20 corresponds to the rider being at least 6 meters above the center, which occurs only when the cosine is at most -1/2, a smaller portion of the cycle than the guide assumed.