A ferris wheel is 20 meters in diameter and boarded from a platform that is 1 meters above the ground. The six o'clock position on the ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 6 minutes. The function h = f(t) gives your height in meters above the ground t minutes after the wheel begins to turn. Write an equation for h = f(t).
Let's use a sine curve, we could use a cosine as well
amplitude is 10, so h = 10sin kt
the period is 6 minutes, so
2?/k = 6 ----> k = ?/3
sofar: h = 10sin (?/3 t)
this has a min of -10, but our min is to be +1, so we have to raise everything up by 11
so far: h = 10sin(?/3 t) + 11
let's see this through one cycle:
t = 0 , h = 0+11 = 11
t = 1.5 , h = 10 + 11 = 21
t = 3 , h = 0 + 11 = 11
t = 4.5 , h = -10+11 = 1
t = 6, h = 11
sketch this curve.
notice our min is at t = 4.5 or at t = -1.5
But we want our min to be at t = 0, so we could move our curve 1.5 to the right, or 4.5 to the left
y = 10 sin ?/3(t - 1.5) + 11
or
y = 10sin ?/3(t + 4.5) + 11
test both equations for t = 0, 1.5, 3, 4.5, and 6
check:
http://www.wolframalpha.com/input/?i=plot+y+%3D+10+sin+(%CF%80%2F3(t+-+1.5))+%2B+11,+y+%3D+10+sin+(%CF%80%2F3(t+%2B+4.5))+%2B+11
(notice that the two curves coincide and are correct, also notice I had to put in extra brackets)
Well, don't worry too much about that equation. If you're afraid of heights, just close your eyes and imagine yourself floating on a fluffy cloud, away from all those numbers and calculations. Ah, the tranquility!
To write an equation for h = f(t), we need to consider the characteristics of the ferris wheel and its movement.
The diameter of the ferris wheel is 20 meters, so its radius (r) will be half of that, which is 10 meters.
At the six o'clock position, the ferris wheel is level with the loading platform, which is 1 meter above the ground. This means that at t=0, the height (h) above the ground will be 10 meters + 1 meter = 11 meters.
The ferris wheel completes 1 full revolution in 6 minutes, which means it completes one full cycle in 6 minutes. Since one complete revolution covers a circumference of 2πr, where r is the radius, in this case, r =10 meters, we know that the ferris wheel completes one full cycle in a distance of 2π(10) = 20π meters.
Now, let's think about how the height of an object changes as it goes around a circle. The height is at its maximum when the object is at the topmost point of the circle (at 12 o'clock position) and at its minimum when the object is at the bottommost point of the circle (at 6 o'clock position).
Since the height at the 6 o'clock position is 11 meters above the ground, we can conclude that the height at the 12 o'clock position will be 11 meters above the height at the 6 o'clock position, which is 11 + 10 = 21 meters.
We can now create an equation for the height above the ground, h, as a function of time, t:
h = 21 sin(2πt/6) + 11
In this equation, t represents the time in minutes, and sin(2πt/6) represents the sine function for an angle equivalent to the fraction of time passed in respect to the full cycle (2πt/6).
Note: This equation assumes that t=0 represents the moment the ferris wheel starts turning, and it assumes a standard sine function for the height.
To write an equation for h = f(t), we need to consider the given information about the ferris wheel.
We know that the diameter of the ferris wheel is 20 meters, which means the radius is half of that, or 10 meters. The loading platform is 1 meter above the ground, so the initial height of the rider is 1 meter.
The ferris wheel completes one full revolution in 6 minutes. This means that it takes 6 minutes for the rider to go around once and return to the same position. At the six o'clock position, the rider is level with the loading platform.
To determine the height of the rider at any given time, we need to consider the position of the ferris wheel as it rotates. The height can be modeled using the sine function, which has a maximum value of the radius and a minimum value of the negative radius.
The general equation for the sine function is: y = A * sin(B(x - C)) + D
Where:
- A represents the amplitude, which is half of the difference between the maximum and minimum height (radius in this case).
- B represents the vertical stretch factor. In this case, B = 2π/period, where the period is the time it takes for one full revolution.
- C represents the horizontal shift or phase shift. In this case, C = 0 because we want the starting point to be at t = 0.
- D represents the vertical shift or the initial height of the rider.
Substituting the given values in the equation, we have:
A = 10 (radius)
B = 2π/6 (one revolution in 6 minutes)
C = 0
D = 1 (initial height)
Therefore, the equation for h = f(t) would be:
h = 10 * sin((2π/6)t) + 1
This equation gives the height above the ground, h, in meters, at time t, in minutes, after the ferris wheel begins to turn.