You are riding the ferris wheel at the Montgomery County Fair. The wheel has a diameter of 36 feet and travels at a constant rate of 3 revolutions per minute. A car at its lowest is 4 feet above the ground. Write a sine function to describe the relationship between time and the height of the car above the ground. State the period, phase shift, vertical shift, and amplitude of the equation.
(i do not understand this question at all, sorry if there isn't any work done)
If you understand since functions, you must understand this. The height of a point on a rotating circle is ha sine (or cosine) function.
The question does not state at what point to start measuring, so we will start with the car at its lowest point at t=0.
Since cos(t) is a max at t=0, we will need to use
h = -Acos(t)
where A is the amplitude. Since the wheel has a diameter of 36, it varies above and below the axle by haf that, or 18.
h = -18cos(t)
But, the car at its lowest is at +4, and not -18, so we need to shift the curve up by 22.
h = 22 - 18cos(t)
But, cos(t) has a period of 2π. cos(kt) has a period of 2π/k. We want a period of 1/3 (3 rpm is the frequency, which is the reciprocal of the period.) So,
2π/k = 1/3 --> k = 6π
h = 22 - 18cos(6πt)
Hmmm. The questions asks for a sine function. Since sin(x) = cos(π/2-x), we finally come to the function
h(t) = 22 - 18sin(π/2 - 6πx)
or
h(t) = 22 + 18sin(6π(x - 1/12))
If you check the graph linked below, you will see indeed that
the amplitude is 18
the period is 1/3
the phase shift is 1/12
http://www.wolframalpha.com/input/?i=y+%3D+22+%2B+18sin%286%CF%80%28x-1%2F12%29%29%2C+y%3D40%2C+y%3D4
To write a sine function to describe the relationship between time and the height of the car above the ground, we need to understand some key concepts.
The general form of a sine function is given by:
y = A * sin(B(x - C)) + D
where:
A represents the amplitude,
B controls the period,
C determines the phase shift, and
D represents the vertical shift.
Let's break down the given information:
1. Diameter of the ferris wheel: The diameter is 36 feet, so the radius (which we'll need for the amplitude) is half of that, 18 feet.
2. Rate of revolution: The wheel completes 3 revolutions per minute. Each revolution corresponds to 2π radians, so in one minute, the wheel covers an angle of 6π radians (3 rev * 2π rad/rev).
3. Lowest point of the car: The car is 4 feet above the ground at its lowest.
Now, let's compute the values for the sine function:
Amplitude (A): The amplitude of a sine function represents half the vertical distance between the maximum and minimum values of the function. In this case, the car reaches its highest point 18 feet above the ground (at the top of the wheel).
Amplitude = (highest point - lowest point) / 2 = (18 - 4) / 2 = 7 feet.
Period (T): The period of a sine function is the length of one complete cycle. It is determined by the rate of revolution.
Since the ferris wheel completes 3 revolutions in one minute, the period is the time taken to complete one revolution, which is 2π/3 radians.
Phase Shift (C): The phase shift of a sine function determines the horizontal displacement of the graph. It tells us where the graph starts. In this case, we'll start at time t = 0, so there is no phase shift. C = 0.
Vertical Shift (D): The vertical shift represents the overall up or down movement of the graph. In this case, the lowest point of the car is 4 feet above the ground, so that is our vertical shift value. D = 4.
Putting all the values together, the sine function that describes the relationship between time (t) and the height of the car (y) above the ground is:
y = 7 * sin((2π/3)t) + 4
Therefore, the period is 2π/3, there is no phase shift (C = 0), the vertical shift is 4 (D = 4), and the amplitude is 7 (A = 7).