[Guest]

A reel is 2 inches in diameter. A thin tape (1/50 of an inch in thickness) is wrapped round the reel until the end of the tape is reached. The diameter of the reel and tape combined is now 4 inches. How long is the tape?

[Guest]

double post

Edited May 15, 2009 by Lemeshianos

[Guest]

There are some interesting ways of "solving" this puzzle posted here. I have noticed that there seems to be an argument on defining some of the aspects of this problem. The way I see it, as a geometry teacher, I would accept at least three correct answers and most likely something in the range of these numbers. The spool has a 2" diameter as a given. Thus, the outer spool surface has a radius of 1".

From here the problem solution may vary.

If you decide to measure the inside face of the tape then you would begin with a radius of 1 and increase the radius by 0.02" which would, of course, change the diameter by 0.04". The tape would be wrapped 50 times ( but your calculations would go from 1 to 1.98 since you are counting the 1) and would give you a series of concentric circles in which you would simply have to find the sum in some fashion, many of you used variations which is fine, to come up with 468.097" as your final length.

If you decide to measure the outer face of the tape then you begin with a radius of 1.02 since this would be the radius after the first wrap. Again, the tape would be wrapped 50 times (your calculations would go from 1.02 to 2) and following the same manner of calculating you would come up with 474.380" as your final answer.

A third way would be to do the previous two and then average them, which would be the same as finding the middle of the tape and measuring that or making your calculations beginning with 1.01 and ending at 1.99. This would give you 471.239 as your final answer.

I would accept any one of these answers as correct, though I would lean towards the latter since it would account for stretch more than the previous two. Again, this is avoiding the irregular increase when joining the first wrap to the second wrap and so on.

[Guest]

A reel is 2 inches in diameter. A thin tape (1/50 of an inch in thickness) is wrapped round the reel until the end of the tape is reached. The diameter of the reel and tape combined is now 4 inches. How long is the tape?

100 feet. I came up with 104.71 feet.

[Guest]

My friends,

The perimeter of a circle is ∏*d not ∏*(r²).

The other thing you need to consider is every time you roll the tape the diameter increases in 2/50 (not 1/50) because increases in both sides... 2/50 = 1/25

Then, my answer is; The length of the tape (L) is the sum of all the "perimeters" around the real from the diameter of 2 to 4 inches.

L= ∏*2 + ∏*(2+1/25) + ∏*(2+2/25) + ... + ∏*(2+50/25) = ∏*(2*50 + (1/25+2/25+ ... + 50/25)) = ∏*(100+((50*51/2)/25)) =

L = ∏*(100+51) = 474.38 in

Hope I was clear!

just a small correction

this line is fine

L= ∏*2 + ∏*(2+1/25) + ∏*(2+2/25) + ... + ∏*(2+50/25) = ∏*(2*50 + (1/25+2/25+ ... + 50/25)) = ∏*(100+((50*51/2)/25)) =

L = ∏*(102+51)

=480 appr

[Guest]

There is no circle

Sorry but why are you calculating the circumference of the tape as if it is a exact circle. Think of a thick tape, say 1 inch thick. Now wrap it around a 50 inches cylinder, you will see that there is gap between the cylinder and the tape where it doubles over itself. Again the outher surface is not a exact circle, if you put it in a cylinder, there will be a gap between cylinder and tape where it ends.

You can not calculate this by circumference method without integral.

Surface method must work here, as somebody proposed before. But they are also wrong because they assumed the tape's sahpe as circle.

The gap between the first roll of the tape and the inner cylinder may be thought as the gap between a circle of 1,02 inches and 1,00 inches while the inner circle touches the outer at a point. There is a semi cresent shape.

My solution:

pi * (2²-1²-(1.02²-1²)/4-(2²-1.98²)/4 * 50= 466 inches.

If you don't get the gap into account the result would be 471 inches.

But my solution is not exact, only more close than others, and I claime you can get it exactly only with integral.

[CaptainEd]

There is no circle

[spoiler=you are all wrong ]Sorry but why are you calculating the circumference of the tape as if it is a exact circle. Think of a thick tape, say 1 inch thick. Now wrap it around a 50 inches cylinder, you will see that there is gap between the cylinder and the tape where it doubles over itself. Again the outher surface is not a exact circle, if you put it in a cylinder, there will be a gap between cylinder and tape where it ends.

You can not calculate this by circumference method without integral.

Surface method must work here, as somebody proposed before. But they are also wrong because they assumed the tape's sahpe as circle.

The gap between the first roll of the tape and the inner cylinder may be thought as the gap between a circle of 1,02 inches and 1,00 inches while the inner circle touches the outer at a point. There is a semi cresent shape.

My solution:

pi * (2²-1²-(1.02²-1²)/4-(2²-1.98²)/4 * 50= 466 inches.

If you don't get the gap into account the result would be 471 inches.

But my solution is not exact, only more close than others, and I claime you can get it exactly only with integral.

Nobody, in the reduction ad absurdum case you propose (imagine a thick tape, like 1 inch), and we follow your accurate advice to recognize that each time around, the tape has to move to a new level and leave a bit of a gap, we have the problem that there is no longer a single radius anyway. Yet our OP gives us "the" radius, as if it is a close approximation.

As long as we're imagining, let's go the opposite direction, imagine an infinitesimally thin tape, like .001 inch. The gap that the tape has to rise above is very small. Might as well just call it zero. If we pretend that it all smooths out, and there is simply a circle (as the OP implies), you can just measure the area, make a long thin rectangle out of it (as I proposed in my post a while back), and calculate the length. Don't need to distinguish whether there are concentric rings 0.02 inch thick or spirals, or even back and forth layers that never actually circumnavigate the spool. Just treat it as oatmeal. How long would this much oatmeal stretch if it were smooshed down to 0.02 in?

In fact, let's go reduction ad absurdum the other way. Let's say I buy your observation that we must calculate the length of a spiral rather than circles. Given that the OP says "the" radius is 1 inch (or it says "the" diameter is 2, I forget what it says), and your observation depends on the fact that it is NOT a perfect circle, and hence has a varying diametric profile, then what diametric profile are YOU assuming for the spool plus tape?

So, I'm not convinced that the entire thread is "all wrong".

Edited May 18, 2009 by CaptainEd

[Guest]

Nobody, in the reduction ad absurdum case you propose (imagine a thick tape, like 1 inch), and we follow your accurate advice to recognize that each time around, the tape has to move to a new level and leave a bit of a gap, we have the problem that there is no longer a single radius anyway. Yet our OP gives us "the" radius, as if it is a close approximation.

As long as we're imagining, let's go the opposite direction, imagine an infinitesimally thin tape, like .001 inch. The gap that the tape has to rise above is very small. Might as well just call it zero. If we pretend that it all smooths out, and there is simply a circle (as the OP implies), you can just measure the area, make a long thin rectangle out of it (as I proposed in my post a while back), and calculate the length. Don't need to distinguish whether there are concentric rings 0.02 inch thick or spirals, or even back and forth layers that never actually circumnavigate the spool. Just treat it as oatmeal. How long would this much oatmeal stretch if it were smooshed down to 0.02 in?

In fact, let's go reduction ad absurdum the other way. Let's say I buy your observation that we must calculate the length of a spiral rather than circles. Given that the OP says "the" radius is 1 inch (or it says "the" diameter is 2, I forget what it says), and your observation depends on the fact that it is NOT a perfect circle, and hence has a varying diametric profile, then what diametric profile are YOU assuming for the spool plus tape?

So, I'm not convinced that the entire thread is "all wrong".

Since this is a tape of given length, and not an engineering problem where we have to account for stretch and imperfections, we can just assume a perfect knife that makes perfect cuts with no loss of tape. The tape lengths can then be laid over one another with no sloping. It doesn't matter how thick or thin the tape is, as long as the thickness is uniform.

[Guest]

Nobody, in the reduction ad absurdum case you propose (imagine a thick tape, like 1 inch), and we follow your accurate advice to recognize that each time around, the tape has to move to a new level and leave a bit of a gap, we have the problem that there is no longer a single radius anyway. Yet our OP gives us "the" radius, as if it is a close approximation.

As long as we're imagining, let's go the opposite direction, imagine an infinitesimally thin tape, like .001 inch. The gap that the tape has to rise above is very small. Might as well just call it zero. If we pretend that it all smooths out, and there is simply a circle (as the OP implies), you can just measure the area, make a long thin rectangle out of it (as I proposed in my post a while back), and calculate the length. Don't need to distinguish whether there are concentric rings 0.02 inch thick or spirals, or even back and forth layers that never actually circumnavigate the spool. Just treat it as oatmeal. How long would this much oatmeal stretch if it were smooshed down to 0.02 in?

In fact, let's go reduction ad absurdum the other way. Let's say I buy your observation that we must calculate the length of a spiral rather than circles. Given that the OP says "the" radius is 1 inch (or it says "the" diameter is 2, I forget what it says), and your observation depends on the fact that it is NOT a perfect circle, and hence has a varying diametric profile, then what diametric profile are YOU assuming for the spool plus tape?

So, I'm not convinced that the entire thread is "all wrong".

I say that this shape is not a circle. If you assume tape's width 0.001 inch, obviously you see it as a circle. But this is just as a blindness as when you look a fault from closer, you realize it and when you look a fault from far, you can't realize it and this time you yourself make a fault.(this is also to chicory from BD)

Really there are no true inner or outer diameters in this shape, thus the question is based on this error. But if you get the outer diameter as the longest diameter(?) of the shape, this makes sense. But still without integral I can't get the exact length of the tape, but suggest my solution a bit closer than yours.

[Guest]

I say that this shape is not a circle. If you assume tape's width 0.001 inch, obviously you see it as a circle. But this is just as a blindness as when you look a fault from closer, you realize it and when you look a fault from far, you can't realize it and this time you yourself make a fault.(this is also to chicory from BD)

Really there are no true inner or outer diameters in this shape, thus the question is based on this error. But if you get the outer diameter as the longest diameter(?) of the shape, this makes sense. But still without integral I can't get the exact length of the tape, but suggest my solution a bit closer than yours.

the area is the integral-- or am I wrong?

[Guest]

the area is the integral-- or am I wrong?

Sorry, I couldn't get what you intended to ask.

Integral gives the area under a curve, defined by a function.

Most simply get a function as y=x, the area under this function is equal to integral of x, which is 1/2 x^2.

Just as in this problem, the area you find by integral, is also equal to the some repeated addition's sum.

[Guest]

i meant that the perimeter of a circle 2pi*r the integral of the perimeter of a circle is the area of a circle. pi*r*r

and how do you get superscripts?

[Guest]

A reel is 2 inches in diameter. A thin tape (1/50 of an inch in thickness) is wrapped round the reel until the end of the tape is reached. The diameter of the reel and tape combined is now 4 inches. How long is the tape?

I haven't read all the posts, but I think there is an easier way to solve this:

Volume of tape:

Its a hollow cylinder...

If the width of the tape is x...

2*pi*4*x - 2*pi*1*x

= 3*pi*x

if the tape is rolled out its volume would be l*w*b

we need to find length...

L*x*1/50

the volumes are equal so:

L*x*1/50 = 3*pi*x

L= 150pi

oops.. some one has already suggested this

Edited May 31, 2009 by adiace