5. A water wheel has a radius of 60 feet, and the center is 14 feet above the water level. The water wheel makes a complete rotation every 45 minutes. A specific cup starts at the top of the wheel. Write a sine equation to model the above water wheel.
Height above water wheel or above water level?
That is a very slow water wheel!
To write a sine equation to model the motion of the water wheel, we need to consider the height of the cup at any given time.
First, let's define the initial position of the cup, which is at the top of the wheel. Since the center of the wheel is 14 feet above the water level, the initial position of the cup is at a height of 14 feet.
Next, let's think about how the cup's height changes as the wheel rotates. We know that the wheel makes a complete rotation every 45 minutes. This means that it completes one full cycle in that time.
The height of the cup can be modeled using a sine function, since sine functions repeat in cycles. The general form of a sine function is:
y = A * sin(Bx - C) + D
Where:
- A represents the amplitude (the maximum height of the function).
- B represents the frequency or period (the number of cycles per unit).
- C represents the phase shift (the horizontal displacement of the function).
- D represents the vertical shift (the midline or average value of the function).
In our case:
- The amplitude is the radius of the wheel, which is 60 feet.
- The frequency or period is the time it takes for the wheel to make a complete rotation, which is 45 minutes.
- The phase shift is 0, since the cup starts at the top of the wheel.
- The vertical shift is the center of the wheel above the water level, which is 14 feet.
Therefore, the sine equation to model the water wheel is:
y = 60 * sin((2π/45)x) + 14