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pudley4
01-28-2005, 12:14 PM
The earlier math/logic question thread referenced an SAT problem from several years ago. Here is the original question (paraphrased, with diagram):

Circle A has a circumference of 4pi.
Circle B has a circumference of pi.

Circle B makes one trip clockwise around Circle A. It rolls the entire way, keeping in contact with Circle A the entire time.

How many revolutions of itself does Circle B make?

http://img.photobucket.com/albums/v208/pudley4/SAT.jpg

Note: This image is not necessarily to scale, so don't take out a ruler and measure it /images/graemlins/wink.gif

Patrick del Poker Grande
01-28-2005, 12:19 PM
We're assuming perfect, no-slip conditions?

Freakin
01-28-2005, 12:23 PM
[ QUOTE ]
We're assuming perfect, no-slip conditions?

[/ QUOTE ]

This isn't rocket science, it's the SATs

Freakin

ThaSaltCracka
01-28-2005, 12:25 PM
6

Reef
01-28-2005, 12:28 PM
The wording is very strange.. Revolutions "of itself" ? If I had to pick, I'd say 4

Piz0wn0reD!!!!!!
01-28-2005, 12:29 PM
4?

ThaSaltCracka
01-28-2005, 12:30 PM
[ QUOTE ]
6

[/ QUOTE ]3?

Piz0wn0reD!!!!!!
01-28-2005, 12:30 PM
[ QUOTE ]
[ QUOTE ]
6

[/ QUOTE ]3?

[/ QUOTE ]
lol!

ThaSaltCracka
01-28-2005, 12:32 PM
[ QUOTE ]
[ QUOTE ]
6

[/ QUOTE ]3?

[/ QUOTE ]5?

elwoodblues
01-28-2005, 12:35 PM
Am I missing something or is this question just asking to see if you know what "circumference" means?
4 (assuming no slippage to help our resident rocket scientist)

pudley4
01-28-2005, 12:40 PM
[ QUOTE ]
The wording is very strange.. Revolutions "of itself" ? If I had to pick, I'd say 4

[/ QUOTE ]

As opposed to "How many revolutions does it make around Circle A (which would be one /images/graemlins/wink.gif )Not sure exactly how they worded it. That's the closest I can come to describing it. Alternate wording:

How many times does Circle B spin completely around?

And yes, no slippage at all. It's math, not physics /images/graemlins/laugh.gif

kyro
01-28-2005, 12:44 PM
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
6

[/ QUOTE ]3?

[/ QUOTE ]5?

[/ QUOTE ]

quite impressive. i think you just got PWNED!...by yourself.

BruinEric
01-28-2005, 01:33 PM
The answer has to be 4 unless I'm mising something. The use of Pi seems curious here. Same answer should apply if you said Circle B has a circumference of X and Circle A of 4X.

If this is the SAT and I'm rushing to get through the questions, this is one I quickly answer 4 and move to the next.

Mano
01-28-2005, 01:35 PM
I believe the correct answer is 5. If you label the initial contact point, start it rolling and see where it contacts the big circle again(so Pi length total contact) you will see that it has rotated by pi/2 (90 degrees) relative to the original circle. It does this 4 times, so we get an extra rotation from the 4 that we would have had if it was rolling on a flat surface of length 4pi.

Patrick del Poker Grande
01-28-2005, 01:36 PM
[ QUOTE ]
I believe the correct answer is 5. If you label the initial contact point, start it rolling and see where it contacts the big circle again(so Pi length total contact) you will see that it has rotated by pi/2 (90 degrees) relative to the original circle. It does this 4 times, so we get an extra rotation from the 4 that we would have had if it was rolling on a flat surface of length 4pi.

[/ QUOTE ]
No.

Mano
01-28-2005, 01:37 PM
Consider if the inner circle had a radius approaching zero. The outer circle would still revolve once due to the circular path.

The Ocho
01-28-2005, 01:38 PM
Wouldn't it rotate around it's center 4 times, but revolve around A once? The wording is confusing.

Mano
01-28-2005, 01:40 PM
If the inner radius is zero, how many times does the outer circle rotate going around it?

elwoodblues
01-28-2005, 01:40 PM
[ QUOTE ]

I believe the correct answer is 5. If you label the initial contact point, start it rolling and see where it contacts the big circle again(so Pi length total contact) you will see that it has rotated by pi/2 (90 degrees) relative to the original circle. It does this 4 times, so we get an extra rotation from the 4 that we would have had if it was rolling on a flat surface of length 4pi.

[/ QUOTE ]

What? If it goes 90 degrees with one rotation, how does that lead you to 5 rotations for 360 degrees?

Mano
01-28-2005, 01:42 PM
It goes an extra 90 degrees (rotates 360 degrees due to the rolling + 90 degrees due to change in orientation).

elwoodblues
01-28-2005, 01:44 PM
I see what you're saying. Much like the moon not revolving per se, but having 1 revolution per year because it travels in orbit around the sun.

junkmail3
01-28-2005, 01:45 PM
cir·cum·fer·ence (sr-kmfr-ns)
n.
The boundary line of a circle.

The boundary line of a figure, area, or object.
Abbr. c or circ. The length of such a boundary .

If one boundry has one unit, and another boundry has 4 ... you can cut it and stretch out the circle so it's flat.

--
--------

It will make four revolutions. I don't think this needs to be overthought.

DemonDeac
01-28-2005, 01:45 PM
the answer is 4
lets look at it this way:
if you were to stretch circle A out into a straight line, it's length would be 4, since that is its diameter (divide 4pi by pi). circle B, for every rotation goes 1 since that is its diameter. thus in order for a 1 diameter circle to travel 4, it must made 4 rotations.

elementary , my friend


of course, if im wrong i look like an idiot

junkmail3
01-28-2005, 01:46 PM
I beat you. So I'll take the idiot lable if we're wrong.

Mano
01-28-2005, 01:48 PM
But it also rotates due to the circular path it takes. If the inner circle had a radius of zero, the outer circle would still rotate once.

junkmail3
01-28-2005, 01:49 PM
[ QUOTE ]
It goes an extra 90 degrees (rotates 360 degrees due to the rolling + 90 degrees due to change in orientation).

[/ QUOTE ]

This doesn't make sense.

Patrick del Poker Grande
01-28-2005, 01:53 PM
[ QUOTE ]
[ QUOTE ]
I believe the correct answer is 5. If you label the initial contact point, start it rolling and see where it contacts the big circle again(so Pi length total contact) you will see that it has rotated by pi/2 (90 degrees) relative to the original circle. It does this 4 times, so we get an extra rotation from the 4 that we would have had if it was rolling on a flat surface of length 4pi.

[/ QUOTE ]
No.

[/ QUOTE ]
Upon further review (thinking for more than 0.2 seconds), this is actually right. End of argument.

Mano
01-28-2005, 01:54 PM
Thank you.

junkmail3
01-28-2005, 01:58 PM
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
I believe the correct answer is 5. If you label the initial contact point, start it rolling and see where it contacts the big circle again(so Pi length total contact) you will see that it has rotated by pi/2 (90 degrees) relative to the original circle. It does this 4 times, so we get an extra rotation from the 4 that we would have had if it was rolling on a flat surface of length 4pi.

[/ QUOTE ]
No.

[/ QUOTE ]
Upon further review (thinking for more than 0.2 seconds), this is actually right. End of argument.

[/ QUOTE ]

I don't like the SATs.

BruinEric
01-28-2005, 01:58 PM
[ QUOTE ]
But it also rotates due to the circular path it takes. If the inner circle had a radius of zero, the outer circle would still rotate once.

[/ QUOTE ]

This plainly can't be true. Let's take the concept of a "radius of zero" (uh..that's a NOTHING, not a circle) out of the equation for a second. Instead, let's say you reduce the circle A in this example to a very small circle.

In fact, let's do this in real life:

Make circle A a marble. Make circle B a basketball. (these measurements are not akin to the initial post, they are an illustration to counter Mano's point.)

If you take circle B and rotate it so it travels the circumference of circle A once, would you also say that circle B (the basketball) does one complete rotation of itself? Nonsense.

Try it yourself. Start at the inflation hole and see if you get back to the inflation hole in one "trip" around the marble. Nope.

If your concept of adjoining rotation as mentioned in this post were true, then how would gears work?

I'm open to the fact that I'm just applying plain logic and missing some advanced thinking, but I just don't see it.

Reef
01-28-2005, 02:02 PM
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
6

[/ QUOTE ]3?

[/ QUOTE ]5?

[/ QUOTE ]

I saw that coming

Justin A
01-28-2005, 02:04 PM
[ QUOTE ]
[ QUOTE ]
But it also rotates due to the circular path it takes. If the inner circle had a radius of zero, the outer circle would still rotate once.

[/ QUOTE ]

This plainly can't be true. Let's take the concept of a "radius of zero" (uh..that's a NOTHING, not a circle) out of the equation for a second. Instead, let's say you reduce the circle A in this example to a very small circle.

In fact, let's do this in real life:

Make circle A a marble. Make circle B a basketball. (these measurements are not akin to the initial post, they are an illustration to counter Mano's point.)

If you take circle B and rotate it so it travels the circumference of circle A once, would you also say that circle B (the basketball) does one complete rotation of itself? Nonsense.

Try it yourself. Start at the inflation hole and see if you get back to the inflation hole in one "trip" around the marble. Nope.

If your concept of adjoining rotation as mentioned in this post were true, then how would gears work?

I'm open to the fact that I'm just applying plain logic and missing some advanced thinking, but I just don't see it.

[/ QUOTE ]


In your basketball around the marble scenario, remember that the marble is stationary and the basketball is rotating aroud it. The basketball will rotate slightly more than 360 degrees in this scenario if you use the backround as a frame of reference.

Rick Diesel
01-28-2005, 02:05 PM
It all depends on how hot the chick rolling circle B is, and in particular how big her boobs are.

mmbt0ne
01-28-2005, 02:22 PM
</font><blockquote><font class="small">En réponse à:</font><hr />
I believe the correct answer is 5. If you label the initial contact point, start it rolling and see where it contacts the big circle again(so Pi length total contact) you will see that it has rotated by pi/2 (90 degrees) relative to the original circle. It does this 4 times, so we get an extra rotation from the 4 that we would have had if it was rolling on a flat surface of length 4pi.

[/ QUOTE ]

Yup. Take 2 quarters, rotate one around the other, it takes 2 rotations even though they have the exact same circumference.

pudley4
01-28-2005, 02:23 PM
in white

<font color="white"> 5 </font>

The student got the answer right and the SAT testers got it wrong (they said <font color="white"> 4 </font> )

Explanations and pictures to follow...

DMBFan23
01-28-2005, 02:24 PM
http://img192.exs.cx/img192/8247/pwn3d.jpg

we all agree that of the big circle were flat, circle B would make one revolution in between frames 1 and 2. that part is just the definition of circumference. but that is taking the surface of the big circle as the frame of reference. if you take the background as the frame of reference, you will see that circle B CANNOT have made only one revolution, because if it had the blue dot would have been facing the same way. you can see from the picture that it's made an extra 1/4th of a revolution, and it does that at each 1/4 of the circle, adding up to one extra revolution, or 5 revolutions.

to think about it in the way that you were, stretch the circle out into a line, and have your circle roll down it. 4 revolutions, right? but now, after each 1/4th of the way down your straight line, bend the line as if bending a paper clip into a square. your little circle rotates, and it's that extra rotation that adds up to 5 rotations.

M2d
01-28-2005, 02:31 PM
aren't SAT's multiple choice?

BruinEric
01-28-2005, 02:43 PM
Awesome. What a conundrum caused by rotation factors when my mind immediately leaps to circumference=distance=same as 2 str8 lines=4. This explains why I did great on SAT Math and didn't do great in Calculus.

Without the time or desire for "second order" thinking, I would have answered this "4" and been on to the next question while "outside the box thinker kid" would still be wrestling over "frame of reference" and not finishing the test in time. Plus, my test gets marked right and his wrong. What fun.

I love standardized tests!

BruinEric
01-28-2005, 02:47 PM
Clearly the problem here is that the question asks about "rotations" which can be interpreted using 'degrees' and 'frame of reference.' One trip from blue dot to blue dot (POINT A) is 1.25 rotations, but really still only one trip on the small circle from START POINT TO END POINT.

If they had asked "see this blue dot here at POINT A" on the small circle? How many times (after starting the trip) does Point A touch the boundary of the larger circle in one complete trip around it. Answer = 4

jimdmcevoy
01-28-2005, 02:49 PM
If someone one is running 5 miles/hour relative to you and he turns on his flashlight, how fast is the light going that is from the flashlight, relative to you?

Assume had he been standing still the speed of the light from his flashight would be x.

Btw, this is the coolest question ever, for nerds.

Patrick del Poker Grande
01-28-2005, 02:49 PM
[ QUOTE ]
Clearly the problem here is that the question asks about "rotations" which can be interpreted using 'degrees' and 'frame of reference.' One trip from blue dot to blue dot (POINT A) is 1.25 rotations, but really still only one trip on the small circle from START POINT TO END POINT.

If they had asked "see this blue dot here at POINT A" on the small circle? How many times (after starting the trip) does Point A touch the boundary of the larger circle in one complete trip around it. Answer = 4

[/ QUOTE ]
It's definitely all in the wording.

pudley4
01-28-2005, 02:53 PM
The SAT testers wanted to test the ability of the student to translate a simple word problem into a simple division problem (4pi/pi = 4). However they didn't take geometry into account /images/graemlins/laugh.gif

Picture 1: Here we have the beginning state. The red dot is our reference point. In each of the subsequent pictures Circle B will have rolled exactly 1 unit (pi) around the bigger circle. This can be verified by remembering the following two facts:

1 - Circle B is exactly 1 unit in circumference. So when it rolls 1 unit in a direction, the same point will again be contacting the larger circle.

2 - The circumference of Circle A is exactly 4 units. So each time Circle B rolls 1 unit, it will move exactly 1/4 of the way around Circle A.

http://img.photobucket.com/albums/v208/pudley4/SAT1.jpg

Picture 2:

http://img.photobucket.com/albums/v208/pudley4/SAT2.jpg

Notice that the red dot is on the left side of Circle B. This means that Circle B has rotated 450 degrees, even though it has only traveled one unit.

Picture 3:

http://img.photobucket.com/albums/v208/pudley4/SAT3.jpg

Now the red dot is at the top of Circle B. Circle B has now made 2 1/2 revolutions while only traveling 2 units.

Picture 4:

http://img.photobucket.com/albums/v208/pudley4/SAT4.jpg

Expected. 3 3/4 revolutions for Circle B now.

Finally we get back to the beginning and Circle B has completed 5 total revolutions.

Congrats to Mano, the only one who answered correctly /images/graemlins/laugh.gif

DMBFan23
01-28-2005, 03:33 PM
subquestion: what if I were holding the light sideways?

subquestion: if I were running at the speed of light holding a mirror, could I see my reflection?

DemonDeac
01-28-2005, 05:15 PM
[ QUOTE ]
Awesome. What a conundrum caused by rotation factors when my mind immediately leaps to circumference=distance=same as 2 str8 lines=4. This explains why I did great on SAT Math and didn't do great in Calculus.


Without the time or desire for "second order" thinking, I would have answered this "4" and been on to the next question while "outside the box thinker kid" would still be wrestling over "frame of reference" and not finishing the test in time. Plus, my test gets marked right and his wrong. What fun.

I love standardized tests!

[/ QUOTE ]




hahahahah, well said

the answer is still 4 in my mind. just look at my explanation. im stickin to it

daryn
01-28-2005, 05:21 PM
</font><blockquote><font class="small">In risposta di:</font><hr />
If someone one is running 5 miles/hour relative to you and he turns on his flashlight, how fast is the light going that is from the flashlight, relative to you?

Assume had he been standing still the speed of the light from his flashight would be x.

Btw, this is the coolest question ever, for nerds.

[/ QUOTE ]

come on man.

M2d
01-28-2005, 05:22 PM
[ QUOTE ]
subquestion: if I were running at the speed of light holding a mirror, could I see my reflection?

[/ QUOTE ]

No, because your optic nerves have mass.

Dr. Strangelove
01-28-2005, 07:14 PM
One thing no one's taking into account is where the question falls on the test, i.e. is it the the 1st or the last question in a session. If it's the 1st this is an easy "4."
If it is the last the answer can't be 4. Since the SAT people said the answer was 4 it must have been one of the first questions in a session and the kid should have known to answer 4 and THEN point out how stupid the question is rather than answer 5 and go through the hassle of getting them to change his 780 to an 800.

scotty34
01-28-2005, 07:45 PM
The answer is still c (speed of light), and not c+5. I don't think this question is for nerds however. I learned this in my grade 11 physics class. Its a very basic part of Einstein's theory of relativity.

jimdmcevoy
01-29-2005, 02:02 AM
correct, but how cool is that? well I think it's cool.