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/
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