91Ó°ÊÓ

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 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.

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 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.