You are standing beside a merry-go-round that your friend is riding. The merry-go-round is 8m in diameter. Describe how the shape of the sine curve models the distance from you and your friend. Identify the function that will model this situation as well as a function that will model the situation when you are standing 4m away from the merry-go-round. In your explanation use the following terms:
Sine
Function
Radius
Repeat
Rotate
Amplitude
Period
Intercept
Maximum
Minimum
Axis of the curve
google circular functions for many discussions, videos, etc.
In this situation, we can use the sine function to model the distance between you and your friend on the merry-go-round. Let's start by considering the shape of the sine curve.
The sine curve is a periodic function that describes oscillatory motion or waves. It takes the form of a smooth, repetitive, and wavy curve. It is defined on a coordinate system where the x-axis represents time or rotation, and the y-axis represents the amplitude or displacement.
Now, let's relate this shape to the distance between you and your friend. Since your friend is riding on a merry-go-round, the distance between you and your friend can be compared to the displacement of a point on the edge of the merry-go-round.
The radius of the merry-go-round, which is half of its diameter, represents the maximum distance between you and your friend. As your friend rotates around the merry-go-round, the distance between you and your friend changes. At certain points, the distance is maximum (when your friend is directly opposite you) or minimum (when your friend is closest to you). These maximum and minimum distances correspond to the maximum and minimum points on the sine curve.
The axis of the sine curve is a horizontal line passing through its center, which represents the average distance between you and your friend when you are both on opposite sides of the merry-go-round. This is the intercept of the sine curve.
Now, let's talk about the mathematical representation of these distances using functions. The general form of the sine function is:
y = A * sin(B * x + C) + D
- A represents the amplitude, which is the maximum distance between you and your friend.
- B represents the frequency or how quickly the merry-go-round rotates. It determines how many cycles of the sine curve occur in a given interval.
- C represents the phase shift or how much the sine curve is shifted horizontally.
- D represents the intercept or average distance between you and your friend.
For the specific situation where you are standing beside the merry-go-round, we can use the sine function as:
y = 4 * sin(x)
In this case, the amplitude is 4 since the diameter of the merry-go-round is 8m, and you are standing half the distance away.
Now, let's consider the situation where you are standing 4m away from the merry-go-round. To model this scenario, we need to shift the sine curve horizontally by 4 units to the right. The function representing this situation becomes:
y = 4 * sin(x - 4)
By subtracting 4 from the x-axis, the sine curve now represents the distance between you and your friend when you are standing 4m away.
In summary, the shape of the sine curve models the distance from you and your friend on the merry-go-round. The amplitude represents the maximum distance, the period represents how often the distance repeats, and the intercept represents the average distance between you both. The sine function with appropriate parameters can be used to model different scenarios, accounting for distance, rotation, and position on the merry-go-round.