I need trig help

Question

Leah 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 12 meters long and the center of the Ferris wheel is 17 meters above the ground. Imagine an angle with a vertex at the center of the Ferris wheel subtending the path Leah travels since the Ferris wheel started rotating. Let θ represent this angle's measure in radians.

1. Write an expression (in terms of θθ) that represents Leah's varying height above the center of the Ferris wheel (in meters).
2. Write an expression (in terms of θθ) that represents Leah'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 problem 1, the starting position at 3:00 would be a height of zero since we are measuring height above center. As the wheel rotates, the rider reaches a maximum height at the "12:00" position which is 90° or π/2 in a standard unit circle. The height could be modeled by a 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 12 which makes it y = 12sin(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) = 12sin[((2π)/P)θ]

The only difference between this equation and the one for problem 2 is the addition of the distance off the ground (17 meters) so the equation for problem 2 would be:

h(θ) = 12sin[((2π)/P)θ] + 17