How Much Carpet is Left?

Date: 03/27/99 at 04:52:15
From: Zene Gillette
Subject: Determining length of rolled goods

We use rolled carpet in our business and would like to know 
how much is left on a roll given the remaining diameter.

Given: Beginning OD, core OD, number of layers, length of roll, New OD

Find:  Length of carpet used or remaining. 

Thanks, Zene

Date: 03/27/99 at 21:12:33
From: Doctor Rick
Subject: Re: Determining length of rolled goods

Hi, Zene, welcome to Ask Dr. Math.

I enjoy solving problems from the working world - students often ask 
us, "Will I ever have a use for this stuff?" The answer is, you'd be 
surprised how often math comes in handy at work.

Here is a method that will give an approximate length of the carpet 
roll; the thinner the carpet relative to the diameter of the core, the 
better the approximation.

Consider a cross-section of the roll. It is an annulus (like a washer 
- a disk with a hole in the middle). Its area is the area of the 
outer circle minus the area of the inner circle (the core).

Let C be the core diameter, F the diameter of a full roll, and L the 
length of carpet on a full roll. Let's also define T, the thickness of 
the carpet.

The cross-section area of the full carpet is 

  Area = (pi/4)(F^2 - C^2)

(Read F^2 as F squared.) When you unroll the carpet, its cross-
sectional area is LT (L times T). This area should be (approximately) 
the same as the area of the roll, so we can find the thickness T:

  T = (pi/4)(F^2 - C^2)/L

You can check the accuracy of the calculation so far by counting the 
layers of carpet on the full roll; this should be 

  n = (F - C)/(2T)
    = (F - C)L/((pi/2)(F^2 - C^2))
    = 2L/(pi(F+C))

Now, if the diameter of a partial roll is D, we can use the same area 
calculations as above, but use the known T to find the length P of 
the carpet on the partial roll:

  P = Area/T
    = (pi/4)(D^2 - C^2)/((pi/4)(F^2 - C^2)/L)
    = L(D^2-C^2)/(F^2-C^2)

So there you have it. The length of carpet on a partial roll is the 
length on a full roll times the difference between the diameter 
squared and the core diameter squared, divided by the same calculation 
for the full roll.

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