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

In this scenario, the shape of the sine curve can accurately model the distance between you and your friend on the merry-go-round. Here's how:

The sine function is a mathematical representation of a wave-like pattern. It is ideal for describing periodic motion or oscillation, which is precisely what happens as your friend rides the merry-go-round.

The input to our function will be the angle at which your friend is positioned on the merry-go-round, while the output will be the distance between you and your friend. To establish this function, we need to consider a few key terms and concepts:

1. Radius: The radius of the merry-go-round is half its diameter, so in this case, it would be 8m/2 = 4m. The radius determines how far your friend can be from the center of the merry-go-round.

2. Repeat: As your friend rotates around the merry-go-round, the distance between you and them repeats itself. This is because the motion is periodic and will continue indefinitely as long as the merry-go-round keeps spinning.

3. Amplitude: The amplitude of the sine curve represents the maximum value it reaches in either direction from its axis. In this case, the amplitude corresponds to the maximum distance you can be from your friend. Since you are standing next to the merry-go-round, the amplitude would be 4m.

4. Period: The period of a sine curve is the length of one complete cycle. In this context, it represents the distance it takes for your friend to complete one full rotation around the merry-go-round. The period of this function will be the same as the circumference of the merry-go-round, given by 2π times the radius. In this case, the period is 2π * 4 = 8π meters.

5. Intercept, Maximum, and Minimum: The intercepts of the curve occur when the distance between you and your friend is zero, which will happen when your friend is directly opposite to you on the merry-go-round. The maximum and minimum values correspond to the furthest and closest distances between you and your friend.

6. Axis of the curve: The axis of the curve is a horizontal line that represents the average value of the function. In this case, it would correspond to the average distance between you and your friend when they are at various angles along the merry-go-round.

Now, to construct the function that models the distance between you and your friend for any angle, we can use the general form of a sine function:

f(x) = A * sin(Bx + C)

Using the information we previously discussed, the specific function that models the distance from you to your friend is:

f(x) = 4 * sin(x)

This function will give you the distance between you and your friend as your friend rotates around the merry-go-round, with an amplitude of 4m and a period of 8π meters.

Additionally, if you want to model the situation when you are standing 4m away from the merry-go-round, you need to shift the function horizontally by 4 units (since the radius is 4m). The modified function would be:

g(x) = 4 * sin(x - π/2)

Here, π/2 is the shift required to position the function correctly when you are standing 4m away from the merry-go-round.

By utilizing these functions, you can accurately describe the shape of the sine curve and how it models the distance between you and your friend on the merry-go-round.