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Length of a Roll of Carpet

Date: 08/21/97 at 15:41:38
From: Shane Massey
Subject: Length of a roll

How would one determine the length of a roll of (say carpet)
without uncoiling it? You are allowed to measure the thickness of the 
carpet and the  diameter of the roll.

I started by letting the thickness of a layer = delta x, but I did not 
make much headway.

Date: 08/22/97 at 11:16:10
From: Doctor Guy
Subject: Re: Length of a roll

Let's say you have a roll of carpet that has an outer radius of 
exactly 12 cm and an inner radius of exactly 4 cm (sort of like a roll 
of paper towels, in that there is an empty space in the middle) and 
the carpet is exactly 1 cm thick. 

The way I would calculate the length of the carpet is this: find the 
area of the region between these two circles (the "washer") and then 
divide this by the thickness of the carpet. The area of the region is 
pi*12^2 - pi*4*4 = pi*(144-16) = 128*pi or about 401.92 

Now if you divide this by the thickness of the carpet, you would get a 
length of about 402 cm (rounding off), or 4 meters (about 12 feet and 
some). The units work here, which is not only nice, but essential.

For more information on how to solve these kinds of problems, as well 
as alternative methods for solving them, have a look at these problems 
from our archives:   

-Doctor Guy,  The Math Forum
 Check out our web site!   
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School Practical Geometry

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