The Ferris wheel at a carnival has a diameter of 18m and descends to 2m above the
ground at its lowest point. Assume that a rider enters a car at this point and rides the wheel
for two revolutions
My main problem is how to get the phase shift. (The answer says its 270 degrees)
well, since sin(x) has a max at x=90, and a min at x=270, that phase shift makes sense. You will have something along the lines of
sin(x+270)
The radius of the wheel is 9, and the whole wheel is lifted 2m, so we have
9sin(x+270)+11
see, at x=0, y=9(-1)+11 = 2
at x=90, y=9(0)+11 = 11
and so on
Thank you very much Steve!
I just want to know how you got 270 in a simpler way please.
Ah, the phase shift! Well, I'm here to help, but I must confess I'm not much of a mathematician. However, I can come up with a joke related to Ferris wheels that might lighten the mood while we figure this out:
Why did the scarecrow hop on the Ferris wheel?
Because he heard it was an "a-maize-ing" ride!
Okay, let's give it a shot. To determine the phase shift, we need to figure out the angular displacement of the Ferris wheel from its lowest point after two complete revolutions. Since one revolution is equal to 360 degrees, two revolutions would be 720 degrees.
Now, assuming the lowest point is at the 0-degree mark, we need to determine how many degrees the wheel has shifted from there. Since the answer claims it's 270 degrees, it means the wheel has shifted three-quarters of a full revolution counterclockwise.
Well, numbers can be tricky, and they like to play tricks on us. So, always double-check the calculations! But remember, even if you get stuck in the math maze, laughter is never far away.
To understand how to calculate the phase shift in this scenario, let's break it down step by step.
1. First, it's important to note that the phase shift represents the horizontal displacement of the starting point of the function. In this case, we need to determine the phase shift of the rider on the Ferris wheel.
2. Let's consider a simple sinusoidal function that models the height of the rider above the ground as a function of time. The general equation for such a function is given by:
h(t) = A * sin(B * (t - C)) + D
Where:
- A represents the amplitude or the maximum height the rider reaches above the ground.
- B represents the frequency or the number of cycles of the function that occur in a given time interval.
- C represents the phase shift or the horizontal displacement of the starting point of the function.
- D represents the vertical shift or the average height above the ground.
3. In this specific scenario, the rider starts at the lowest point, which is 2m above the ground. Hence, the vertical shift (D) is equal to 2.
4. The rider goes through two full revolutions on the Ferris wheel. This means the function will repeat twice within the time interval.
5. Since the diameter of the Ferris wheel is 18m, the distance covered by the rider in one complete revolution is equal to the circumference of the wheel, which is πd = π * 18m.
6. Since the rider goes through two complete revolutions, the entire time interval for the function is 2 times the time it takes to complete one revolution. Let's call this time interval T.
7. To find the frequency (B) of the function, we need to divide the number of cycles in the interval (2) by the time interval (T). So B = 2 / T.
8. The phase shift (C) can be calculated by determining the horizontal displacement of the starting point. In this case, since the starting point is at the lowest point (2m above the ground), the phase shift can be calculated as the fraction of one complete revolution needed to reach this point. This fraction can be calculated as (C / 360°) = (2m / 18m).
9. Now we can calculate the phase shift (C) by rearranging the equation from Step 8: C = (360° * 2m) / 18m.
10. Simplifying the equation from Step 9, we find that C ≈ 40°.
11. However, the phase shift is usually expressed in degrees or radians within one complete cycle. To convert the degrees of the phase shift into the degrees within one complete cycle (360°), subtract the calculated phase shift from 360°: C' = 360° - 40°.
12. Simplifying the equation from Step 11, we find that C' ≈ 320°.
Therefore, the phase shift in this scenario is approximately 320 degrees (C' ≈ 320°). It seems that the answer you mentioned (270 degrees) may be incorrect, or there might be additional information or considerations that I am unaware of.