I need help with this trigonometry problem

Question

Raquel boards a Ferris wheel at the 3-o'clock position and rides the Ferris wheel for one full rotation. The radius of the Ferris wheel is 10 meters long and the center of the Ferris wheel is 16 meters above the ground. Imagine an angle with a vertex at the center of the Ferris wheel subtending the path Raquel travels since the Ferris wheel started rotating. Let θ represent this angle's measure in radians.

A. Write an expression (in terms of θ) that represents Raquel's varying height above the center of the Ferris wheel

B. Write an expression (in terms of θ) that represents Raquel's varying height above the ground (in meters).

Answer 1

We are not told which direction the Ferris wheel rotates but I'll assume its a standard counterclockwise direction. Here is a sketch:

For part A, the starting position at 3:00 would be a height of zero since we are measuring height above center. As the wheel rotates, the height of the rider reaches a maximum at 12:00 which is 90° or π/2 from the starting position. This is the standard "sine" function and so we can use y = sin(θ) as our base equation. We would need to multiply by the amplitude which is 10 which makes it y = 10sin(θ). We are also not told how fast the wheel rotates. So we can introduce a variable into the equation to cover that. We can use "P" for period (the time it takes to go all the way around from 3:00 to 3:00). We can also entitle the function "h" for height. The function then would be:

h(θ) = 10sin[((2π)/P)θ]

The only difference between this equation and the one for part B is the addition of the distance off the ground (16 meters) so the equation for part B would be:

h(θ) = 10sin[((2π)/P)θ] + 16