Finding the Length of Carpet Left on a Roll
Date: 09/25/2008 at 20:08:45 From: Lee Subject: Can you find the total square yardage of a roll of carpet I can see the end of a roll of carpet and figure out the approximate square yardage by taking the inside circumference, outside circumference and number of total layers left. I need to see if there is a formula. EX: 12"=inside roll circumference, 24"=outside roll circumference, 12 layers on end of roll. The difference between 24" and 12" is 12", so each layer must be about 1" more than the previous layer (i.e. first layer length = 12", 2nd = 13", 3rd = 14" . . . 24th = 24"). Now, when I add all layers together I get a certain total length. What is the equation for such a problem?
Date: 09/25/2008 at 23:41:06 From: Doctor Peterson Subject: Re: Can you find the total square yardage of a roll of carpet We have many answers to this kind of question; you might try searching our archives for the phrase "roll of carpet" to find those that happen to be about carpets. Here is one, with references to others: Determining Length of Material Remaining on a Roll http://www.mathforum.org/library/drmath/view/67031.html My formula there is L = pi/4 (Do^2 - Di^2)/t where Do = outer diameter Di = inner diameter t = thickness of one layer You give the circumferences, which we'll have to divide by pi, and since you know the number of layers, we can make a different formula based on a number of layers N. Let's say we have Co = outer circumference [so Do = Co/pi] Ci = core circumference [so Di = Ci/pi] N = number of layers [so t = (Do-Di)/(2N) = (Co-Ci)/(2 pi N)] Then my formula becomes L = pi/4 (Co^2 - Ci^2)/(pi^2) * (2 pi N)/(Co - Ci) = (Co + Ci)*N/2 For your numbers this gives L = (24 + 12)*12/2 = 216 inches This is exactly what I would get as an approximation using your idea of the linear increase in circumference with each layer. The sum of an arithmetic series like your 12 + 13 + 14 + ... + 23 (note that you counted 13 layers by including both 12 and 24) is the average of the first and last, times the number of terms. In your example, this is 12*(12+23)/2 = 210 inches. But what about the last layer? We only got to a length of 23. If we split the difference and treat the series as 12.5 + 13.5 + ... + 23.5, our length turns out to be 12*(12.5+23.5)/2 = 216. And in fact, we could derive the formula above starting with this idea of the arithmetic series. Actually, I suspect that in the case of carpet it is possible that the backing would be more rigid than the top, so that the actual length would be better estimated by the 210 inch figure; my 216 assumes that the outside of each layer expands and the inside contracts equally when it is rolled up. The formula using your idea turns out to be L = (Co + Ci - (Co - Ci)/N)*N/2 = ((N-1)Co + (N+1)Ci)/2 I've never looked at it from this perspective before and obtained this particular formula! It's not quite as simple, and considering that the carpet really forms a spiral rather than neat layers, it's still just an approximation; but it has an interesting "elegance"! If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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