Problem 3 - College Trigonometry

Question

The London Eye is a popular tourist attraction in London, England and is one of the largest Ferris Wheels in the world. It has a diameter of 135 meters and makes one revolution (counterclockwise) every 30 minutes. It is constructed so that the lowest part of the Eye reaches ground level, enabling passengers to simply walk on to, and off of, the ride. Find a sinsuoid which models the height \(h\) of the passenger above the ground in meters \(t\) minutes after they board the Eye at ground level.

Step-by-Step Solution

Answer

The sinusoid that models the height is \( h(t) = 67.5 \sin \left( \frac{\pi}{15}t \right) + 67.5. \)

Step 1: Understand the Constraints of the Problem

The London Eye has a diameter of 135 meters, which means its radius is half of that: \( \, r = \frac{135}{2} = 67.5 \, \) meters. Since passengers board at ground level, the lowest point of the ferris wheel is 0 meters above the ground. The center of the ferris wheel, then, is 67.5 meters above the ground. The wheel makes one complete revolution every 30 minutes.

Step 2: Determine the Characteristics of the Sine Function

The height function \( h(t) \) is a sinusoidal function because it repeats every 30 minutes. The height varies from 0 meters to 135 meters, thus the amplitude \( A \) is half of the total variation: \( A = 67.5 \) meters. The vertical shift (midline) is also 67.5 meters, which is the radius of the wheel.

Step 3: Define the Sine Function's Argument

The sine function completes one cycle in 30 minutes, so the period \( T \) is 30. The standard sine function \( \sin(x) \) has a period of \( 2\pi \), so the argument of the sine function must be adjusted: \( \omega = \frac{2\pi}{30} = \frac{\pi}{15} \). Therefore, the argument of the sine function in terms of \( t \) is \( \omega t = \frac{\pi}{15}t \).

Step 4: Construct the Sinusoidal Model

Given the characteristics identified, the height \( h(t) \) above ground is modelled by the transformed sine function: \[ h(t) = 67.5 \sin \left( \frac{\pi}{15}t \right) + 67.5. \]Here, 67.5 is the amplitude and the vertical shift, while \( \frac{\pi}{15}t \) accounts for the adjusted period of 30 minutes.

Key Concepts

Ferris Wheel Model

A Ferris Wheel provides a fascinating real-life application of a sinusoidal function. When you think about the motion of a Ferris wheel, it's all about a smooth, repetitive circle. On the London Eye, passengers experience such movement as they revolve at a consistent rate.
Since the Ferris Wheel moves in a circular motion, math can describe the passenger's height above the ground as a sinusoidal function. This sinusoidal pattern captures the rise and fall that passengers feel on the wheel.
The importance of this lies in being able to model situations that are predictable over time, like the revolution of the wheel. When the passengers begin their ride at the lowest point, they start at zero height from the ground. As the wheel turns, they move upwards, reach a peak (the top of the wheel), and then descend again. This consistent pattern is what makes the Ferris Wheel a perfect example of sinusoidal modeling.

• Diameter of 135 meters, radius is 67.5 meters.
• The lowest point is ground level, 0 meters.
• Circumference of the wheel is complete in 30 minutes.

Understanding this model helps to interpret other natural phenomena that follow a repetitive cycle over time.

Amplitude and Period in Trigonometry

In trigonometry, the sinusoidal function is crucial for modeling periodic phenomena due to its defining characteristics of amplitude and period.
Amplitude is the measure from the middle of the wave to its peak or trough. It tells us the maximum extent of vibration or oscillation measured from the position of equilibrium. For the London Eye, this is half of its diameter, so the amplitude is 67.5 meters.
The period refers to the time it takes for one complete cycle of the wave to pass a fixed point. It determines how long it takes for the cycle to repeat itself. In the case of the London Eye example, this is the 30-minute journey for a full revolution.

• Amplitude is 67.5 meters, reflecting the radius of the wheel.
• Period is 30 minutes, indicating a complete rotation.

These parameters allow us to place values into mathematical functions to create models. Understanding them can illuminate a wide variety of naturally occurring cycles, from the movement of celestial bodies to biological rhythms.

Height Function Modeling

The Height Function Modeling utilizes a combination of trigonometric principles to predict the height of a passenger at any given time as they ride the Ferris Wheel.
The Ferris Wheel's motion is best described by a sine function, which is ideal for modeling periodic, wave-like patterns. For the London Eye, the model equation is:
\[ h(t) = 67.5 \sin \left( \frac{\pi}{15}t \right) + 67.5. \]
This function encapsulates both the circular movement of the wheel and the passenger's journey as they revolve.
Key elements of this model include:

• Amplitude of 67.5 meters, representing the maximum height deviation from the center.
• A vertical shift (or midline) fixed at 67.5 meters, the center of the Ferris wheel above the ground.
• Adjusted period given by the argument \( \frac{\pi}{15}t \), matching the 30-minute revolution.

Using this function, you can calculate how far above the ground a passenger will be at any time, providing insights into their entire ride experience. It's a testament to how mathematics helps us represent and understand predictable natural processes.