Trig Experts and people who know Pre-Calc...help!?

Question

The wheel of a ferris wheel has a 20 m diameter and turns at 4 rpm. Its lowest point is 1.3 m above the ground. Assume that the height (h) is a sinusoidal function of time (t) where t=0 is the lowest point.

a) Write a equation for h.
Answer:
-Okay, so I went through my book and found the equation h=a*sin*w*t+k (I have absolutely no idea what a or w stand for though.)
-Then, I plugged in the numbers and got h=10*sin*4*t+11.3

Is that right? Do you have any idea how to find the equation or explain what I'm doing?

b) Find h at t=10sec.
Answer:
-I figure I would just plug in 10 for t, but how do I solve h=10*sin*40+11.3

Please help! I have pretty much no idea what I'm doing, and I need to understand it for a test. I posted as far as I got, and you don't have to give me the answer. Just tell me what I'm doing...please ?!?

Answer 1

Hi,

First, determine how long it takes for one revolution of the Ferris wheel. Since it makes 4 revolutions per minute, it makes a revolution every 15 seconds. So the speed of the ferris wheel is15 seconds/ revolution.

Next, determine the minimum height of the ferris wheel. This is the beginning point of the path of travel at time 0 seconds at the bottom of the wheel. So, your starting point is at (time, min ht.) or
( 0 sec., 1.3 meters).

The maximum height the ferris wheel reaches will be after the time for half a revolution, 7.5 seconds, and 20 meters higher than the lowest point at ( 7.5 sec., 21.3 meters).

Now write the location of the rider after one complete revolution as ( seconds, height). This would be ( 15 sec., 1.3 meters). This would be the second minimum point of the graph.

To model the curve produced by the path of the ferris wheel using the sine function, use the equation:
y = a sin( b( x - c)) + d where
a = amplitude (vertical stretch of the graph)
b = number of cycles data completes over the course of the natural period of the function. If using radians, 2Pi/ b = Period. If using degrees, 360/ b =Period. Solving these for b, 2Pi/Period = b in radians or 360/ Period = b in degrees.
c = horizontal phase shift of data
d = vertical shift of graph (center line)

The amplitude, "a", is found by finding the difference of the maximum and minimum values, divided by 2. In your case this is also the radius of the ferris wheel. Either way, ( 21.3 - 1.3)/2 or 20/2 both equal 10. This is the value of the amplitude, a.

To find "b", use the formula 2Pi/Period = b in radians or 360/ Period = b in degrees. That means 2Pi/15 = b in radians or 360/15 = 24 = b in degrees.

"c" represents the horizontal shift of the graph to the left or the right. In the case of the ferris wheel, the horizontal shift when using the sine function is the time it takes the rider to reach the center line of the graph. Since this would require the Ferris wheel to move through 1/4 of a rotation, it would take 1/4 of 15 seconds or 3.75 seconds.

"d" is the vertical shift of the graph to the height of the new center line. It would be the amplitude plus the minimum value of the graph. In this case, it is 1.3 + 10 or 11.3

So the equation you would get using radians would be:
y = 10 sin( (2Pi/15)( x - 3.75)) + 11.3

The equation you would get using degrees would be:
y = 10 sin( 24( x - 3.75)) + 11.3

For part b, to find the height of the person on the Ferris wheel after 10 seconds, let x = 10 in the above equation. Your answer should be 16.3 feet.

I hope this helps you.

Answer 2

From the information given it's unclear to me what you are doing, but I can help you out with some of it.

Equation given:
h = aSin(wt + k).

a - Amplitude. This is the maximum height form the base line.
w - Omega. Revolution per unit of time. (2 π f).
k - lead or lag value.

So a = 20/2 = 10.
w = 4 rpm.
k = ? This is unclear, maybe it should be 1∙3

After ten seconds:
h = aSin(wt + k).
h = 10 Sin((4)(10) + 1∙3)
h = 10 Sin(41∙3) [Assuming degrees].
h = 10(0∙66000 1668..)
h = 6∙6000 1668..
h ≈ 6∙6 m

Answer 3

a)Since they have set the lowest point at t=0, the best trig function to use would be cosine. Since cosine usually starts at the top of the amplitude and go downwards, you need a negative cosine function. The equation for h would be:
h= -10 cos(4t) + 1.3
The negative refers to the shape of the graph (up to top, then down back to original height), the 4t refers to time period, the +1.3m at the end refers to its original height (which is 1.3m off the ground). I think this is right, but forgive me if i'm wrong since I finished this stuff quite a while ago.

b) To find the answer, just put in 1/6 min for t, since ur t value is based on minutes, not seconds.

Answer 4

Out of the three, this is how it goes, easiest to hardest. 1. Stats (easy, as long as you can read graphs and stuff) 2. Trig (you focus on mainly one thing, which is pretty easy) 3. Precalc is actually harder than trig. It's harder than calculus, too. In precalc, you learn A LOT of stuff which prepares you for calculus (which is why it is harder than calculus, because by the time you get there, you know everything already). That's what most people say. I've taken all, and I have to agree, too. I'd go with trig.

Answer 5

a is the amplitude which is half the difference of the highest point (21.3 m) minus the lowest point (1.3 m) 21.3 - 1.3 = 20; half that is 10.

Answer 6

"w" (it's actually an omega) is "angular frequency", measured in radians per second. remember that one rotation is 2*pi radians, so your 4 rpm is 8pi radians per minute, or (2/15)pi radians per second. this is your w value.