A ferris wheel is 20 meters in diameter and boarded from a platform that is 3 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.
What is the Amplitude?
What is the Midline?
What is the Period?
How High are you off of the ground after 3 minutes?
f(0) is a minimum, so you want something like
h = M - Acos(kt)
The radius of the wheel is 20-3
The axle is at h=20, so now we know that
h = 20 - 17cos(kt)
The period is 6, so 2π/k = 6
h = 20 - 17cos(π/3 t)
to find h(3) you can do the math, but you know that it is 1/2 revolution, so it is at max height: 20+17 = 37
Well, well, well, let's ride this word problem, shall we?
Amplitude is like the wheel's party animal friend - it represents just how much the wheel likes to get high, or in this case, how high it goes above the midline. Since the ferris wheel's diameter is 20 meters, and the radius is half of that, the amplitude is equal to half the diameter which is... drumroll, please... 10 meters!
Ah, the midline, the sweet spot where the ferris wheel hangs out most of the time. Since the loading platform is 3 meters above the ground, that's the midline right there. Simple, right?
Next up - the period, or how long it takes for the wheel to complete one full revolution. We're told it takes 6 minutes to do that, so the period is 6 minutes. Easy peasy!
Now, after 3 minutes, we want to know how high we are off the ground. To find that out, we can plug in t = 3 into the function. So, h = f(3) = ...hm?
Oh, wait, I almost forgot to give you the actual function. Silly me. The function h = f(t) gives your height above the ground, right? Well, since the wheel is 10 meters above the midline when it's at its highest point, and the midline is 3 meters above the ground, we can write the function as h = 10sin(t) + 3. Got it now?
So, h = f(3) = 10sin(3) + 3. Pop that into your calculator, and voila! You'll find out exactly how high you are off the ground after 3 minutes. Enjoy the view, my friend!
To find the amplitude, midline, and period of the function h = f(t), we can analyze the given information about the ferris wheel.
The amplitude of a periodic function is half the distance between the maximum and minimum values. Since the ferris wheel has a diameter of 20 meters, the radius (and hence, the amplitude) can be calculated as half of the diameter: amplitude = 20 / 2 = 10 meters.
The midline of a periodic function is the horizontal line that represents the average value of the function. In the case of the ferris wheel, the midline corresponds to the height of the loading platform, which is 3 meters above the ground. Therefore, the midline is at a height of 3 meters.
The period of a periodic function is the time it takes for the function to repeat itself. In this case, the ferris wheel completes 1 full revolution in 6 minutes, which means it takes 6 minutes for the height function to complete a full cycle. Therefore, the period is 6 minutes.
To find how high you are off the ground after 3 minutes, you can plug t = 3 into the function h = f(t):
h = f(3)
h = 10 * sin((2π/6) * 3) + 3
h = 13 meters
Therefore, after 3 minutes, you are 13 meters above the ground.
To answer these questions, we need to understand the general form of a sinusoidal function, which is given by:
h = A*sin(B(t - C)) + D
Where:
A represents the amplitude,
B represents the number of cycles (or periods) per unit of time,
C represents the horizontal shift (phase shift),
D represents the vertical shift (midline).
Now let's apply this information to the given problem.
1. Amplitude:
The amplitude of a sinusoidal function is the maximum distance from the midline. In this case, the ferris wheel has a diameter of 20 meters, so the radius (and amplitude) would be half of that, which is 10 meters. Therefore, the amplitude (A) is 10.
2. Midline:
The midline represents the average value of the function. In this case, the wheel is boarded from a platform 3 meters above the ground. So the midline (D) would be 3 meters.
3. Period:
The period of a sinusoidal function is the time it takes for one complete cycle. Given that the wheel completes 1 full revolution in 6 minutes, the period (P) would be 6 minutes.
4. Height after 3 minutes:
To find the height after 3 minutes, we need to substitute t=3 into the function.
h = 10*sin((2π/6)*(3 - C)) + 3
To find the value of C, we know that the six o'clock position on the ferris wheel is level with the loading platform, which means that at t=0, the height is 3 meters. Substituting t=0 into the equation gives us:
3 = 10*sin((2π/6)*(0 - C)) + 3
Simplifying, we get:
0 = 10*sin(-Cπ/3)
Since sin(0) = 0, we can know that -Cπ/3 = 0, which means C = 0.
Substituting t=3 and C=0 into the equation:
h = 10*sin((2π/6)*3) + 3
Simplifying, we get:
h = 10*sin(π) + 3
Since sin(π) = 0, the height at t=3 is:
h = 0 + 3 = 3 meters.
So, after 3 minutes, you are 3 meters off the ground.
To summarize:
- The amplitude is 10 meters.
- The midline is 3 meters.
- The period is 6 minutes.
- After 3 minutes, you are 3 meters off the ground.