Problem 37 - Precalculus Mathematics for Calculus

Question

Ferris Wheel A ferris wheel has a radius of \(10 \mathrm{m},\) and the bottom of the wheel passes \(1 \mathrm{m}\) above the ground. If the ferris wheel makes one complete revolution every \(20 \mathrm{s}\), find an equation that gives the height above the ground of a person on the ferris wheel as a function of time.

Step-by-Step Solution

Answer

\( h(t) = 10 \sin \left( \frac{\pi}{10} t - \frac{\pi}{2} \right) + 11 \)

Step 1: Understanding the Ferris Wheel Parameters

The ferris wheel has a radius of 10 m, meaning the center is 10 m above the lowest point. Because the bottom of the wheel is 1 m above the ground, the center of the ferris wheel will be 11 m above the ground (1 m + 10 m).

Step 2: Determining the Period

Since the ferris wheel makes one complete revolution every 20 seconds, the period of the circular motion is 20 seconds.

Step 3: Using the Sine Function for Height Variation

The height of a point on the ferris wheel above the center follows a sine pattern. At time zero, the height should be at its bottommost point, 1 meter above the ground. Thus, the sine function needs a phase shift. The general form for the height function is \( h(t) = A \sin(Bt + C) + D \).

Step 4: Amplitude and Vertical Shift

The amplitude \( A \) is the radius of the wheel, 10 meters, because the tallest to shortest height above the center varies by 20 meters (double the radius). The vertical shift \( D \) is the center height, 11 meters above the ground.

Step 5: Finding the Phase Shift and Frequency

To account for the sine function starting at the lowest point, we use \( \sin(\theta - \frac{\pi}{2}) \), equivalent to \(-\cos(\theta) \). The frequency is found with \( B = \frac{2\pi}{T} = \frac{2\pi}{20} = \frac{\pi}{10} \).

Step 6: Putting It All Together

Substitute the values into the equation: \( h(t) = 10 \sin \left( \frac{\pi}{10} t - \frac{\pi}{2} \right) + 11 \). The shift \( -\frac{\pi}{2} \) in the argument converts the sine function to start from the lowest point.

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry that describes periodic oscillations. Understanding it is crucial for describing various scenarios in physics, such as circular motion and wave behavior.
This function \(y = \sin(\theta)\) oscillates between -1 and 1, creating a wave-like pattern. When applied to circular motion, the sine function represents how a particular point moves vertically around a circle as time progresses.

• The function \(y = A \sin(Bt + C) + D\) modifies the basic sine function to model real-world phenomena by adjusting amplitude, frequency, phase shift, and vertical shift.

In the ferris wheel problem, the sine function models a person's height above the ground as they move circularly on the wheel.

Amplitude and Phase Shift

Amplitude and phase shift are two key parameters in adjusting the sine function for modeling purposes. The amplitude represents the maximum extent of oscillation from the central position.
For a ferris wheel, the amplitude is synonymous with the wheel's radius, which dictates how high and low above the center a rider moves.

• The amplitude \(A\) determines how "tall" or "short" the oscillation is, reflecting the wheel’s radius.
• The phase shift, represented by \(C\) in the equation \(y = A \sin(Bt + C) + D\), horizontally shifts the sine curve along the x-axis. This accounts for shifts in the oscillation start point.

In our exercise, the phase shift helps the function start with the person at the wheel's lowest point, 1 meter above the ground.

Circular Motion

Circular motion describes how an object moves around a fixed point in a circular path. In physics, understanding circular motion is imperative for analyzing objects rotating around a central axis, such as a ferris wheel.
The radius of a circle, representing half the distance across the circle, plays a big role in determining the object's position at any point in time. Using the sine function in our example captures the vertical component of this motion accurately.

• Each full revolution corresponds to one complete cycle of the sine wave.
• The time taken for one complete revolution determines the period of the motion.

For the ferris wheel, this circular motion is completed every 20 seconds, setting the period and influencing the equation parameters.

Mathematical Modeling

Mathematical modeling uses mathematics to describe and predict real-world phenomena. Through equations and functions, complex systems simplify into understandable formats.
By modeling the ferris wheel scenario, one can predict any rider’s exact height at any given time during a ride.

• The equation \(h(t) = 10 \sin \left(\frac{\pi}{10} t - \frac{\pi}{2} \right) + 11\) efficiently models this system.
• Each mathematical parameter (amplitude, phase shift, period) represents a tangible feature of the wheel.

This model helps visualize the entire ride’s ups and downs, giving insight into periodic motions encountered in physics and engineering.