Question

Colossus Added to Six Flags St. Louis in \(1986,\) the Colossus is a giant Ferris wheel. Its diameter is 165 feet; it rotates at a rate of about 1.6 revolutions per minute; and the bottom of the wheel is 15 feet above the ground. Find a function that relates a rider's height \(h\) above the ground at time \(t\). Assume the passenger begins the ride at the bottom of the wheel.

Short answer

The height function is \[ h(t) = 97.5 + 82.5 \cos \left( 3.2 \pi t - \frac{\text{pi}}{2} \right). \]

Step-by-step solution

- Identify the components of the Ferris wheel

The first thing to note is the diameter of the Ferris wheel, which is 165 feet. This means the radius is half of the diameter, so the radius is \(\frac{165}{2} = 82.5\) feet. The bottom of the wheel is 15 feet above the ground.

- Determine the vertical displacement

Let's denote the vertical displacement of a passenger above the center of the Ferris wheel as \(d \). The total height of the rider will be this vertical displacement plus the height of the center of the wheel above the ground, which is the sum of the radius and the bottom's height: \(\text{Center's height} = 82.5 + 15 = 97.5\) feet.

- Find the angular position function

The Ferris wheel rotates at a rate of 1.6 revolutions per minute. Therefore, the angular speed in radians per minute is \(\theta(t) = 1.6 \times 2\text{pi} t = 3.2 \text{pi} t\), where \(t\) is the time in minutes.

- Set up the height function

The height \(h\) of the rider above the ground as a function of time can be modeled using a cosine function since the motion is circular. The general height function will be: \(\text{Height} = 97.5 + 82.5 \cos(3.2 \frac{\text{pi}}{\text{rev}} t)\).

- Adjust for starting position

Since the passenger starts at the bottom of the Ferris wheel, the \(\text{cos}\) function must be shifted. Initially at \(t = 0\), \(\text{Height} = 15\), thus we use \(\theta(t) = - \frac{\text{pi}}{2} + 3.2 \frac{\text{pi}}{\text{rev}} t\).

- Simplify to obtain the final function

Finally, combine all the pieces into one equation: \[ h(t) = 97.5 + 82.5 \cos \left( 3.2 \pi t - \frac{\text{pi}}{2} \right). \]

Key concepts

Trigonometric Functions

Trigonometric functions are fundamental in modeling periodic phenomena, such as the motion of a rider on a Ferris wheel. In this case, we're using the cosine function to describe the height of a rider over time.

The cosine function is particularly useful because it starts at its maximum value when the angle is zero, and it smoothly transitions through all other values in its cycle. This makes it ideal for describing the smooth circular motion of a Ferris wheel.

To understand how we derive the height function, you need to recognize that the height varies sinusoidally (in waves) as the wheel spins. Importantly, the parameters we use in our cosine function (amplitude, period, phase shift) closely relate to the dimensions and speed of the Ferris wheel.

Angular Speed

Angular speed refers to how quickly an object is rotating and is measured in radians per unit time. In this exercise, the Ferris wheel's angular speed is given in revolutions per minute.

To convert it to a more mathematical form, we use the fact that one revolution is equal to \(2\pi\) radians. Thus, if the wheel rotates at 1.6 revolutions per minute, we can express this as:

• \(1.6 \times 2\pi = 3.2\pi\) radians per minute.

This angular speed tells us how fast the angle in our trigonometric functions changes over time. When plugged into our height equation, it ensures that the rider’s height is accurately modeled throughout the wheel's rotation.

Cosine Function

The cosine function is pivotal in this height calculation due to its properties and shape. Given by \( \text{cos}(\theta) \), its values oscillate between -1 and 1.

Here, we adjust the standard cosine function to match the Ferris wheel's specifics. The formula

• \( h(t) = 97.5 + 82.5 \cos (3.2 \pi t - \frac{\pi}{2}) \)

helps us map a rider's height. Key adjustments in the formula include:

• Amplitude: The 82.5 feet represent the amplitude – the maximum extent of vertical displacement from the center.
• Vertical Shift: The 97.5 term shifts the entire cosine curve upwards so that the lowest point is 15 feet above ground.
• Phase Shift: The \( - \frac{\pi}{2} \) adjusts the phase of the wave to account for the starting position.

These adjustments ensure our height function accurately represents the motion of the rider from start to end.

Circular Motion

Circular motion involves any motion that makes a circular path at a constant distance from a center point. In the context of our Ferris wheel problem, the rider's circular motion means their height varies in a predictable way over time.

Grasping the nature of circular motion helps understand the importance of trigonometric functions in modeling it. As the wheel turns, every point on its circumference, including where the rider sits, moves up and down in a smooth wave-like motion.

This is precisely why we use cosine. Its periodic nature aligns well with the continuous, repeating nature of circular motion. By tracking the angular position and relating it to vertical displacement through cosine, we effectively map how the rider’s height changes dynamically.