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### Roll of Paper

```Date: 06/15/2003 at 00:23:39
From: Scotty
Subject: How many feet in a roll of paper 50" high?

I am getting a paper rewinder that runs 6,000 ft a minute, and the
roll is 50" high above the floor. How many miles and feet are there in
this roll of paper and how long will it take to run?
```

```
Date: 06/15/2003 at 01:06:56
From: Doctor Jeremiah
Subject: Re: How many feet in a roll of paper 50" high?

Hi Scotty,

You need to know how big the hole in the center of the roll is and
how thick the paper is.  If you can measure these two things,
calculating the length is easy. But without these two things the
answer would be more of a guess than anything else.

- Doctor Jeremiah, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 06/15/2003 at 21:53:00
From: Scotty
Subject: How many feet in a roll of paper 50" high?

The roll core is 3 inches in diameter, and the thickness of the paper
is 30# newsprint; I believe it's 30 thousandths thick (very thin
paper.)
```

```
Date: 06/16/2003 at 10:54:26
From: Doctor Jeremiah
Subject: Re: How many feet in a roll of paper 50" high?

Hi Scotty,

The whole roll is 50" diameter (25" radius) and the inner core is 3"
diameter (1.5" radius) so the total thickness of the paper is 23.5".
The paper is 30/1000" (0.03") thick so the number of layers of paper
is 23.5"/0.03" = 783 1/3, which is the same as 2350/3.  Now, the roll
might not be exactly 50" and if it is even one layer thicker that
will change the answer. Feel free to measure the outside and inside
diameters to an accuracy of 30 thousandths if you are concerned.

None of this really helps because each layer is longer than the last
layer because it's farther away from the center. Each layer has a
length of 2(Pi)r where r is the distance from the center.  But the
distance from the center is 1.47+.03n where n is the layer number
(starting at n=1)  We had to use 1.47" so that when n was equal to 1
the first layer would be at he 1.5" mark.

So the actual length of each layer is 2(Pi)(1.47+0.03n) and the total
length of the roll is the sum of 2(Pi)(1.47+0.03n) from n=1 to
n=2350/3.  In mathematics:

n=2350/3
+----
\
/    2(Pi)(1.47+0.03n)
+----
n=1

But how do we evaluate that?  Well, first we need to expand the
parenthesis:

n=2350/3
+----
\
/    2(Pi)1.47+2(Pi)0.03n
+----
n=1

And then expand the summation:

n=2350/3          n=2350/3
+----             +----
\                 \
/    2(Pi)1.47 +  /    2(Pi)0.03n
+----             +----
n=1               n=1

The constants do not affect the summation and may be applied afterward
so:

n=2350/3            n=2350/3
+----               +----
\                   \
2(Pi)1.47 /    1 +  2(Pi)0.03 /    n
+----               +----
n=1                 n=1

Now the sum of 1 from 1 to something is n and the sum of n from 1 to
something is n(n+1)/2.  I could prove this to you if you like (just
write back).  If we substitute those results:

2(Pi)1.47(n) +  2(Pi)0.03(n+1)n/2 where n=2350/3

2(Pi)1.47(2350/3) +  2(Pi)0.03(2350/3+1)(2350/3)/2

7235.1 + 57905.3

65140.4 inches

The total roll then is 65140.4 inches or 5428.4 feet. If the roller
does 6000"/minute then the whole roll will take just over 54 seconds
(which is quite fast!)  5428.4 feet is 1 mile and 148.4 feet.

Let me know if you have any other questions.

- Doctor Jeremiah, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 06/18/2003 at 01:29:16
From: Scotty
Subject: How many feet in a roll of paper 50" high?

Roll width is 30 inches, by 40 O.D. roll core is 3 inches, so
subtract 3 inches from the center of the roll. Let's say I'm running
at 5,000 ft per minute. How do I figure the miles in the roll, and
the length of time to run? I'm talking about rolls of paper that go
from us to the printing presses.

Scott.
```

```
Date: 06/18/2003 at 10:25:42
From: Doctor Jeremiah
Subject: Re: How many feet in a roll of paper 50" high?

If the thickness of the paper is known, then you do  the same thing
as in the last example, the main difference being that this roll has
a 40-inch diameter and the other one had a 50-inch diameter (unless
that was a typo)

The paper itself is (40-3)/2 = 18.5 inches thick.  If each layer is
30/1000 inches thick then there are 1850/3 layers (616 2/3).

If we do the same summation we did before:

2(Pi)(inner_diameter-thickness)(#_layers)
+  2(Pi)(thickness)(#_layers+1)(#_layers)/2

We will get the length of the roll.

- Doctor Jeremiah, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Higher-Dimensional Geometry
High School Higher-Dimensional Geometry

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