a ferris wheel with a radius of 14m makes one revolution every 30 seconds. the bottom of the wheel is 2.5m above the ground.
a) find the equation
b) how does the equation change if the graph starts at the centre of rotation
or, since the min occurs at t=0,
h = 9.5 - 7cos((π/15)t)
a) To find the equation of the ferris wheel, you can use a sinusoidal function. A sinusoidal function is given by the equation y = A sin(Bx - C) + D, where A, B, C, and D are constants that control the amplitude, frequency, phase shift, and vertical shift respectively.
In this case, the radius of the ferris wheel is 14m, which means the amplitude (A) is equal to 14. Since the ferris wheel makes one revolution every 30 seconds, the period (T) is equal to 30 seconds. The frequency (f) is the reciprocal of the period, so f = 1/T = 1/30.
The general equation for the ferris wheel can now be written as y = 14 sin(2π/30 x - C) + D. To determine the values of C and D, we need to consider the height of the bottom of the wheel. Given that the bottom of the wheel is 2.5m above the ground, the vertical shift (D) is 2.5.
b) If the graph starts at the center of rotation, the phase shift (C) in the equation will change. Normally, the ferris wheel starts at its lowest point when x = 0, so there is no phase shift. However, if the graph starts at the center of rotation, there will be a phase shift of π/2.
Hence, the equation of the ferris wheel with the graph starting at the center of rotation would be y = 14 sin(2π/30 x - π/2) + 2.5.
To find the equation of the ferris wheel, we can start by observing the key components of the problem:
1. The ferris wheel has a radius of 14m. This means that the distance from the center of the wheel to any point on the circumference is 14m.
2. The ferris wheel makes one revolution every 30 seconds. This implies that it completes a full circle in 30 seconds.
3. The bottom of the wheel is positioned 2.5m above the ground.
a) To find the equation that represents the height of a passenger on the ferris wheel as a function of time, we can use a trigonometric function. Specifically, we can use the sine or cosine function, since they oscillate between -1 and 1.
Let's use the sine function to represent the equation, where h(t) is the height of the passenger at time t:
h(t) = A + B * sin(C * t + D)
In this equation:
- A represents the vertical offset or the average height of the wheel, which is 2.5m above the ground.
- B represents the amplitude or the maximum distance the passenger moves up and down from the average height, which is the radius of the wheel, 14m.
- C represents the frequency or the number of complete cycles in a given time period. In this case, the ferris wheel completes one revolution in 30 seconds, so the frequency is 1/30.
- D represents the phase shift, which is the initial position of the graph. In this case, since we want the graph to start at the bottommost point, we need a phase shift of pi/2 (equivalent to a quarter of the cycle).
So, the equation for the height of the passenger becomes:
h(t) = 2.5 + 14 * sin((1/30) * t + pi/2)
b) If the graph starts at the center of rotation, the equation will change based on the new phase shift.
Since the graph starts at the center of rotation, the height at that point will be the average height of the wheel, which is 2.5m. Therefore, the equation becomes:
h(t) = 2.5 + 14 * sin((1/30) * t)
Now the graph will start at the center of rotation and oscillate up and down from there.
I will assume you want the initial position to be 2.5 m high
Let's start with the basic curve.
amplitude = 7,
period = 2π/k
2π/k = 30
k = π/15
height = 7sin((π/15)t) , where t is in seconds
make a sketch of this to see that at ..
t=0, h = 0
t = 7.5, h = 7sin(π/2) = 7
t = 15, h = 0
t = 22.5 , h = -7
we want our min to happen when t = 0, so move the curve 7.5 to the right
(or 22.5 to the left)
y = 7sin((π/15)(t-7.5) )
but when t = 0, we want that min of -7 to be 2.5
final curve :
h = 7sin((π/15)(t-7.5) ) + 9.5
adjust my equation for b)