## Thursday, January 13, 2011

### Which One Is Your Carpet?

Post-Holidays sales are here. You come to a home decoration store and splurge on a new carpet for your living room. When you drive to pick up the carpet at the back of the store, you see 3 different carpet rolls packed for shipping. Which one is yours? They all appear to have exactly the same type of carpet, all tightly rolled, and only vary by the size of the roll (base of the carpet cylinder). Can you figure out the length of the carpet from the diameter of the roll and size of the little empty hole in the center?

Answers accepted all day long on Friday, on our Family Puzzle Marathon. They will be hidden until Saturday morning (EST) and everyone who contributed something reasonable will get a puzzle point.

Tom said...

I'm not going to load the carpet by myself, and the employee probably knows which is mine. Anyway...

I can come "quite close" to figuring the length of a carpet roll. I still may not know which is mine, of course.

The outer diameter is not enough information.

I'd count the number of "rings" (layers, like tree rings). Let's say it is 18 rings. I'd measure the diameter of the outer ring, say 18". I'd do the same with the inner ring, say 6". (Ignore the diameter of the cardboard spool, if there is one.) I do not need to measure every ring, I think; the average diameter here is 12", and the gains in circumference will be linear with the gains in diameter.

12"d x pi x 18rings = 678 linear inches of carpet, or 56.5 linear feet, or just about 19 linear yards. For the roll I just measured, that is. If the roll is 3 yards in "width" then it contains about 56.5 square yards. If that's what I paid for, then this particular roll is PROBABLY mine.

It'd be "the hard way" to measure each ring and multiply by pi, add them up, etc. It might be more accurate.

I'm trying to recall the last time I bought carpet. I think they sell it by the square yardage of the room they measured. They deliver some extra if they're smart, because there is always waste.

Donna said...

You would have to know the thickness or depth of your carpet. If you look at a carpet from the side, and look at it as a rectangle with length the length of your carpet and width the thickness of the carpet, then the area of that rectangle would be length of carpet x thickness. If we were to roll that rectangle into a circle, the area won't change (maybe slightly due to the crushing of the plush carpet top), but if we can calculate the area of this circle and subtract out the area of the hole, then we should still get approximately the same area. Therefore, we should be able to then divide the area by the thickness or depth of the carpet and find the length.

To do this, you would measure the diameter of the entire roll of carpet, and use the area of a circle formula, pi x r squared, where r is half the diameter. Then measure the diameter of the hole and calculate the area of the hole in the same way. Subtracting the area of the hole from the area of the large carpet-circle would yield the area of the side of the carpet. Now you need to divide out the thickness of the carpet, and that will give you the length. It will be approximate due to the thickness being somewhat reduced when rolled tightly, but will be close enough for you to tell which carpet is your's. I actually tried this with my hallway runner just to verify that it does work, and it was close....within a couple inches.

Maria said...

Wow, people. This is fantastic. Two very different approaches.

Tom's insight that circumference is changing linearly as diameter of the carpet circles grows (Circumference = Pi x Diameter) is brilliant and allows him to use the average diameter. Then, just multiply this average circumference by the # of circles inside the roll.

Donna's idea of using area of the carpet cross-section and the fact that this area barely changes when carpet is rolled is incredibly creative. Just equalize the area of the rectangle (Carpet length x thickness) to the difference between areas of the circles that represent outer and inner edges of the roll.
And she even tried it at home!

Two well-deserved puzzle points.