A Ferris wheel of diameter 18.5m rotated at a rate of 0.2rad/s. If passengers board the lowest car at the height of 3 m above the ground, determine a sine function that models the height, h, in metres, the car relative to the group as a function of the time, t, in seconds
radius = 9.25
how long to go 2 pi radians?
.3rad/sec * T = 2 * 3.14
T = 20.9 seconds for a revolution
height = 3 + 9.25 at center = 12.25
so if it is at the low point at t = 0
h = 12.25 + 9.25 sin (2 pi t/T - pi/2)
the pi/4 is to make it sin (-pi/2) = -1 at t = 0 so it is at 3 m at t = 0
Typo
the pi/2 is to make it sin (-pi/2) = -1 at t = 0 so it is at 3 m at t = 0
To model the height, h, of the car relative to the ground as a function of time, we can use a sine function.
The general equation for a sine function is:
h = A sin(Bt + C) + D
Where:
A: amplitude of the function (half the difference between the highest and lowest points)
B: frequency of the function (determines how quickly the function oscillates)
C: phase shift of the function (horizontal shift)
D: vertical shift of the function
In this case, we know that the diameter of the Ferris wheel is 18.5m, which means the radius is half of that (9.25m). Since passengers board the lowest car at a height of 3m above the ground, the amplitude, A, can be calculated as:
A = (9.25m - 3m) = 6.25m
The frequency, B, can be determined by the rate of rotation, which is 0.2 rad/s. The frequency is the reciprocal of the period, so:
Period (T) = 2π / B
0.2 = 2π / B
B = 2π / 0.2 = 10π rad/s
The phase shift, C, is 0 since the initial position is at the lowest point.
The vertical shift, D, is 3m since passengers board the lowest car at a height of 3m above the ground.
Putting it all together, the sine function that models the height, h, as a function of time, t, is:
h = 6.25 sin(10πt) + 3
To determine a sine function that models the height of the car relative to the ground as a function of time, we need to consider the equation of a sine function and relate it to the given information.
The general equation of a sine function is:
h = A * sin(B * t + C) + D
where:
- A represents the amplitude of the function
- B represents the frequency (or time period) of the function
- C represents any horizontal phase shift
- D represents any vertical shift
In this case, the diameter of the Ferris wheel is 18.5m, so the radius would be half of that, which is 9.25m. Hence, the amplitude (A) of the function would be 9.25m. The frequency (B) is given as 0.2rad/s.
The initial height of the car above the ground when passengers board is 3m, so there is a vertical shift (D) of +3m in the sine function.
Since the Ferris wheel is initially at its lowest point when t = 0, there is a phase shift (C) of -π/2 radians.
Putting it all together, the sine function that models the height (h) of the car relative to the ground as a function of time (t) is:
h = 9.25 * sin(0.2 * t - π/2) + 3
This function will give you the height of the car relative to the ground at any given time.