A Ferris wheel has a diameter of 60 feet. When you start at the bottom of the Ferris wheel, you
are 2 feet from the ground. The Ferris wheel completes one rotation in 2 minutes.
Create a function that represents your height relative to the center of the Ferris wheel as a
function of time.
*in this question i understand that the diameter is 60 so the center as radius is 30 , but i don't understand how to make it as a function time
Now create a function that represents your height from the ground as a function of time.
* i don't understand this one neither PLEEAASE HEEELP
you have to be familiar with the meaning of the constants in an equation such as
y = a sin k(Ø+d) + c
a is the amplitude, our range of heights is 60 ft
so a = 30
y = 30sin kt + c , where t is in minutes
the lowest value so far would be -30, but we need our lowest to be +2
so c = 32
y = 30 sin k(t+d) + 32
the period of our wheel is 2 minutes, so
2π/k = 2
2k = 2π
k = π
so we have y = 30 sin π(t+d) + 32
we now have to "move" our basic curve d units so that
when t = 0 , we have y = 2
when t = 1/2, we have y = 32
when t = 1 , we have y = 62 ***
when t = 3/2, we have y = 32
when t = 2, we have y = 2
using ***
62 = 30sin π(1+d) + 32
sin π(1+d) = 1
I know sin π/2 = 1
so π(1+d) = π/2
1+d = 1/2 --------> d = -1/2
looks like we have:
y = 30 sin π(t - 1/2) + 32
testing:
when t = 0, y = 30sin -π/2 + 32 = 2 , that's good
when t = 1/2, y = 30sin π(1/2-1/2) + 32 = 32, ok
when t = 1 , y = 30 sin π(1/2) + 32 = 62
looks like we got it!!!
To create a function that represents your height relative to the center of the Ferris wheel as a function of time, you can use the equation of a cosine curve.
The equation of a cosine function is given by:
y = A * cos(B * (x - C)) + D
where:
- A is the amplitude of the function (the maximum value)
- B determines the frequency (the number of cycles per unit)
- C is a phase shift (horizontal shift)
- D is the vertical shift
In this case, the amplitude is equal to half the diameter of the Ferris wheel, which is 30 feet. The frequency is given by the number of rotations per unit of time, which is 1 rotation every 2 minutes or 1/2 rotation per minute. The phase shift is 0 because you start at the bottom of the Ferris wheel, and the vertical shift is the distance from the ground to the center of the Ferris wheel, which is 2 feet.
Therefore, the height relative to the center of the Ferris wheel as a function of time can be represented by the equation:
y = 30 * cos((1/2) * t) + 2
where t is the time in minutes.
To create a function that represents your height from the ground as a function of time, you can just subtract the radius (30 feet) from the height relative to the center.
So the height from the ground as a function of time can be represented by the equation:
y = 30 * cos((1/2) * t) - 28
where t is the time in minutes.
To create a function that represents your height relative to the center of the Ferris wheel as a function of time, we can use the sine function. The height can be modeled using the equation:
h(t) = A * sin(B * t)
Where:
- h(t) represents the height of the person relative to the center of the Ferris wheel at time t.
- A represents the amplitude of the function, which is half of the diameter of the Ferris wheel (A = 30 feet).
- B represents the angular frequency of the function, which is calculated as 2π divided by the time it takes for one rotation (B = 2π/2 = π).
So, the function that represents your height relative to the center of the Ferris wheel as a function of time is:
h(t) = 30 * sin(π * t)
To create a function that represents your height from the ground as a function of time, we just need to add the initial height of 2 feet. So the function becomes:
h(t) = 30 * sin(π * t) + 2
This function will give you the height from the ground as a function of time.