AP Precalculus Practice Test: Unit 3 Question #1 Height of a Ferris Wheel Periodic Word Problem

My AP Precalculus Practice Tests are carefully designed to help students build confidence for in-class assessments, support their work on AP Classroom assignments, and thoroughly prepare them for the AP Precalculus exam in May.

The **AP Precalculus Practice Test Unit 3, Question #1** often includes a **periodic word problem** about the height of a passenger on a Ferris wheel. This problem is designed to test your understanding of **periodic functions** (sine and cosine), amplitude, midline, period, and phase shifts. Here's a typical setup and explanation:

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### Problem Description

A Ferris wheel has a diameter of \(60\) feet and makes one complete rotation every \(2\) minutes. The lowest point of the wheel is \(5\) feet above the ground, and passengers board at the lowest point.

1. Write an equation for the height of a passenger above the ground as a function of time, \(t\), in minutes, assuming the ride starts at \(t = 0\) with the passenger at the lowest point.
2. Use the equation to find the height of the passenger at \(t = 1.5\) minutes.

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### Key Features of the Problem

1. **Amplitude**: The amplitude of the periodic function represents half the diameter of the Ferris wheel, which is \(30\) feet.
2. **Midline**: The midline represents the average height of the passenger, which is the radius (\(30\) feet) plus the distance from the ground to the bottom of the Ferris wheel (\(5\) feet). This gives \(35\) feet.
3. **Period**: Since the wheel takes \(2\) minutes to complete one rotation, the period is \(2\) minutes.
4. **Phase Shift**: The problem specifies that the passenger starts at the lowest point, meaning the function is a cosine function reflected vertically.

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### Step 1: Writing the Equation

The general form of a cosine function is:

\[
h(t) = A \cos(Bt + C) + D
\]

- \(A = -30\) (negative because the passenger starts at the lowest point).
- \(B = \frac{2\pi}{\text{Period}} = \pi\) (since the period is \(2\) minutes).
- \(C = 0\) (no horizontal shift because the ride starts at the lowest point at \(t = 0\)).
- \(D = 35\) (midline).

Thus, the height function is:

\[
h(t) = -30 \cos(\pi t) + 35
\]

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### Step 2: Finding the Height at \(t = 1.5\) Minutes

Substitute \(t = 1.5\) into the equation:

\[
h(1.5) = -30 \cos(\pi \cdot 1.5) + 35
\]

\[
h(1.5) = -30 \cos(1.5\pi) + 35
\]

\[
h(1.5) = -30 \cdot 0 + 35 = 35
\]

At \(t = 1.5\) minutes, the passenger's height is **35 feet**.

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### Skills Assessed

This question assesses the following AP Precalculus skills:
1. Identifying properties of periodic functions (amplitude, midline, period, phase shift).
2. Translating a real-world scenario into a trigonometric model.
3. Evaluating trigonometric functions at specific points.

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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