Practice a College Board-style free response question on a sinusoidal Ferris wheel model. Write your response, then reveal the model answer to see exactly what earns each point.
Free Response Question · Unit 3 · Sinusoidal Modeling
A Ferris wheel has a radius of 20 meters, and its center is 25 meters above the ground. It completes one full revolution every 40 seconds. A rider boards at the lowest point at time t = 0 seconds. The rider's height above the ground, in meters, is modeled by h(t) = 25 − 20·cos((π/20)·t), where t is measured in seconds.
t (seconds)
h(t) (meters)
0
5
10
25
20
45
30
25
A
Identify the amplitude, the midline, and the period of h, and interpret each in the context of the Ferris wheel.
✓ Model answer (earns the point)
The amplitude is 20 meters — the coefficient of the cosine — which is the radius of the wheel, the distance the rider rises above and falls below the center. The midline is h = 25 meters, the height of the center of the wheel. The period is 2π ÷ (π/20) = 40 seconds, the time for one full revolution.
Why it scores: Correctly identifies all three values (20, 25, 40 with units) AND interprets each in context — amplitude as the radius, midline as the center height, period as the time per revolution. A value without its contextual meaning would be incomplete.
B
Determine the maximum height the rider reaches and the first time t > 0 at which it occurs. Justify your answer using the structure of the model.
✓ Model answer (earns the point)
The maximum height is 45 meters (midline 25 plus amplitude 20). The height is greatest when −cos((π/20)t) is greatest, which happens when cos((π/20)t) = −1, that is when (π/20)t = π, so t = 20 seconds. The rider first reaches the top of the wheel, 45 meters, after 20 seconds — half of one revolution.
Why it scores: States the maximum (45 m) using midline + amplitude, AND finds the first time by setting the cosine to −1 and solving. Just reading 45 from the table without justifying the time, or vice versa, would lose a point.
C
Find all times t during the first full revolution (0 ≤ t ≤ 40) at which the rider is exactly 35 meters above the ground. Show the steps that lead to your answer.
✓ Model answer (earns the point)
Set h(t) = 35: 25 − 20·cos((π/20)t) = 35, so −20·cos((π/20)t) = 10, giving cos((π/20)t) = −1/2. On the interval 0 ≤ (π/20)t ≤ 2π, cosine equals −1/2 at angles 2π/3 and 4π/3. Solving (π/20)t = 2π/3 gives t = 40/3 ≈ 13.3 seconds, and (π/20)t = 4π/3 gives t = 80/3 ≈ 26.7 seconds. So the rider is 35 meters high at about 13.3 s and 26.7 s.
Why it scores: Isolates the cosine, recognizes that periodicity gives two solutions in one revolution, solves both, and reports them with units. Giving only one solution — a common error — would not earn full credit because the question asks for all times in the interval.
How to score points on AP Precalculus FRQs
Interpret every parameter in context. Amplitude, midline, and period each mean something physical — say what.
Use the structure of the model. Maxima and minima occur where the sine or cosine equals ±1; set the trig part to that value instead of guessing.
Account for periodicity. A trig equation usually has more than one solution in a given interval — find all of them.
Work in radians and show the angle steps. Solve for the angle first, then convert back to t; answers without supporting work may not earn full credit.
Include units. Heights in meters, times in seconds — a bare number can cost a point.