You are standing at a base of a Ferris Wheel which is 4 m above ground while your friend is riding. The Ferris Wheel is 8m in diameter. Describe how the shape of the sine curve models the distance your friend is to the platform you are on. Identify the function that will model this situation as well as a function that will model the if we measure his distance to the ground. In your explanation use the following terms:
Sine
Function
Radius
Repeat
Rotate
Amplitude
Period
Intercept
Maximum
Minimum
Axis of the curve
clearly the axle is 8m above the ground, and the radius is 4m. So, you can start out with a function that describes the distance to the ground. It will look something like
h(t) = 4+4sin(t)
See where you can go from there. I assume the platform is at the wheel's lowest point.
In this situation, the shape of the sine curve is used to model the distance your friend is from the platform you are on as the Ferris Wheel rotates.
The function that models this situation is a trigonometric function, specifically a sine function. The sine function relates the angle of rotation to the vertical distance of your friend from the platform.
The radius of the Ferris Wheel is the distance from the center of the wheel to the outer edge, which is half of the diameter. In this case, the radius is 4m.
As the Ferris Wheel rotates, your friend repeats their motion, going up and down in a periodic manner. This periodicity is represented by the sine curve.
The sine function has an amplitude, which is the maximum distance away from the axis of the curve. In this case, the amplitude is 4m, which is the radius of the Ferris Wheel.
The period of the sine function is the time it takes for the curve to repeat itself. In this case, the period of the function is equal to the time it takes for the Ferris Wheel to complete one full rotation.
The curve intersects the x-axis (horizontal axis) at certain points. These intercepts represent the times when your friend is at the same level as the platform you are on.
The maximum value of the sine function represents the highest point your friend reaches above the platform, while the minimum value represents the lowest point your friend reaches below the platform. The vertical distance between these two points is equal to twice the amplitude.
In summary, the shape of the sine curve models the vertical distance between your friend and the platform you are on as the Ferris Wheel rotates. The function that models this situation is a sine function with an amplitude equal to the radius of the Ferris Wheel, a period equal to the time it takes for one full rotation, and intercepts that indicate when your friend is at the same level as the platform.
To describe how the shape of the sine curve models the distance your friend is to the platform you are on, let's first establish the key terms involved in this situation:
1. Sine: The sine function, denoted as sin(x), is a mathematical function that relates the angles of a right triangle to the ratios of two sides.
2. Function: In mathematics, a function is a relationship that maps each input value (independent variable) to a unique output value (dependent variable). In this case, we are looking for a function that relates the angle of rotation of the Ferris Wheel to the distance of your friend from the platform.
3. Radius: The radius is the distance from the center of the Ferris Wheel to any point on its circumference. In this case, the radius is half the diameter, which is 8m/2 = 4m.
4. Repeat/Rotate: The Ferris Wheel completes a full rotation, or one revolution, as it goes through 360 degrees or 2π radians.
5. Amplitude: The amplitude of a sine function is the maximum distance the function deviates from its average value (or the midline). In this case, the amplitude would be half the diameter of the Ferris Wheel, which is 4m.
6. Period: The period of a sine function is the distance along the x-axis for one complete cycle of the function. In this case, the period would be 360 degrees or 2π radians since the Ferris Wheel completes one revolution.
7. Intercept: An intercept is the point(s) where a graph intersects or crosses an axis. In this case, we are interested in the y-intercepts.
8. Maximum: The maximum point on a sine function is the highest point it reaches on the y-axis, also known as the peak.
9. Minimum: The minimum point on a sine function is the lowest point it reaches on the y-axis, also known as the valley.
10. Axis of the curve: The axis of the curve is the horizontal line that represents the midline of the sine curve. It is the average value of the function.
Now, let's identify the functions that will model this situation:
To model the distance between your friend and the platform you are on, we can use the standard equation for a sine function:
f(x) = A * sin(B(x - C)) + D
Using the given information, we have:
Amplitude (A): The maximum distance your friend will be from the platform is the radius of the Ferris Wheel, which is 4m.
Period (B): The Ferris Wheel completes one revolution for an angle of rotation of 360 degrees or 2π radians. So B = 2π/360 = π/180 radians per degree.
Phase Shift (C): Since your friend is at the bottom of the Ferris Wheel at the start, there is no phase shift, so C = 0.
Vertical Shift (D): The vertical shift is the average distance your friend is from the platform, which is the radius of the Ferris Wheel, 4m.
Therefore, the function that models the distance between your friend and the platform is:
f(x) = 4 * sin((π/180) * x)
To model the distance between your friend and the ground, we need to consider the height of the Ferris Wheel platform. The platform is 4m above the ground, so we need to subtract this distance from the function f(x) to shift it downward by 4m. The function becomes:
g(x) = 4 * sin((π/180) * x) - 4
This function will model the distance your friend is from the ground as a function of the angle of rotation of the Ferris Wheel.
Clearly you want somebody to do this assignment for you.
You have shown no work of your own, nor have you shown us where your difficulty is.
From the data, the only information given would be the amplitude of the sine function and perhaps a vertical shift.