Question

Corey boards a Ferris wheel at the 3-o'clock position and rides the Ferris wheel for several rotations. The Ferris wheel has a radius of 10 meters, and when Corey boards the Ferris wheel he is 15 meters above the ground. Imagine an angle with its vertex at the center of the Ferris wheel that subtends the path Corey travels.

Define a function g that expresses Corey's distance above the ground (in meters) in terms of the number of radians the angle has swept out since the ride started. g(0) =

Answer 1

Corey's height above the ground as he rides the Ferris wheel can be modeled by the function g(θ) = 15 + 10 sin(θ), with the initial height being 15 meters.

Step-by-step explanation:

To define a function g that expresses Corey's distance above the ground in terms of the number of radians θ the angle has swept out since the ride started, we consider the motion of Corey on the Ferris wheel as sine wave oscillation, due to the circular motion.

At the 3-o'clock position, Corey is 15 meters above the ground and directly at the edge of the wheel's circle. Since the radius of the Ferris wheel is 10 meters, the center of the wheel is at a height of 10 meters (the radius) + 5 meters (the difference between Corey's starting height and the radius) for a total of 15 meters above the ground.

The function g(θ) = 15 + 10 sin(θ) models the vertical position of Corey above the ground in meters. This is because Corey's height varies as a sine function of the angle θ. Here, the amplitude is 10 (the radius of the Ferris wheel), and the vertical shift is 15 (the center's height above the ground). The function g(0) is the initial height of Corey above the ground, so g(0) = 15 + 10 sin(0) = 15 meters.