pacomartin
pacomartin
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February 10th, 2013 at 8:00:18 PM permalink
A rug merchant is hired to carpet a hallway formed by two circular concentric walls. Not knowing the radii of the circles he has his men carry in a 20' stick to measure the circumference. Unfortunately when he gets it in the door and closes the door it is completely wedged between the outside wall (at the tips) and the inside wall. Even though he is unable to use the stick, he still knows how much carpet he needs.

How much carpet does he need?
boymimbo
boymimbo
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February 10th, 2013 at 8:31:57 PM permalink
To be clear two circular concetric walls is essentially a circular hallway.


I assume that the hallway is 20' thick. The amount of carpet required is

Let r be the radius of the outside circle.

The area of carpet is pi (r^2) - pi (r-20)^2 = pi(r^2 + 40r -400 - r^2 ) = pi(40r-400)
pi r^2 - pi (r-20)^2 = pi (-400 + 40r) where r is the radius of the outside wall.

I can't see how this can be solved without not knowing r. r > 20.

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CrystalMath
CrystalMath
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February 10th, 2013 at 8:45:11 PM permalink
314.1593 sq ft
I heart Crystal Math.
skrbornevrymin
skrbornevrymin
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February 10th, 2013 at 8:46:36 PM permalink
Edit: recomputing....misunderstood the question the first time.

Quick answer: enough to cover the floor. ha ha.
boymimbo
boymimbo
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February 10th, 2013 at 8:55:02 PM permalink
Crystal, how do you make that assumption?
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skrbornevrymin
skrbornevrymin
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February 10th, 2013 at 9:09:02 PM permalink
Quote: skrbornevrymin

recomputing....misunderstood the question the first time.

Quick answer: enough to cover the floor. ha ha.



Ok. After having thought about this a little, I have determined that this cannot be solved with the information given, because we do not have the length of the hallway, only the width of it. At the very least it would be a circular room with the inner radius of 0' and the outer radius of 20 feet which is what I initially assumed. However, it could be a very long hallway such as that which would circle a basketball arena, for example, which was 20 feet wide.

Regardless of the area of the floor however, carpet comes off of a roll in rectangles (or squares) in most cases, so waste would be a factor as well. So until we get more info, I am sticking with my "quick answer" as stated above.
stargazer
stargazer
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February 10th, 2013 at 9:19:03 PM permalink
Crystalmath is correct. The area of the outer circle minus the area of the inner circle will always be equal to the area of a circle with a radius of 10'
AlanMendelson
AlanMendelson
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February 10th, 2013 at 9:21:50 PM permalink
Oh for heaven's sake. It's a carpet salesman. If he sent a stick that's 20 feet long he will tell you that you need 20 + 20 = 40 X 40 = 1600 square feet since carpet is sold in rectangular rolls, with a "liitle extra just in case you spill something later."
Nareed
Nareed
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February 10th, 2013 at 9:57:11 PM permalink
Quote: AlanMendelson

Oh for heaven's sake. It's a carpet salesman. If he sent a stick that's 20 feet long he will tell you that you need 20 + 20 = 40 X 40 = 1600 square feet since carpet is sold in rectangular rolls, with a "liitle extra just in case you spill something later."




Of course. But if these problems happened to a real salesman, he'd use a tape measure rather than a stick.
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skrbornevrymin
skrbornevrymin
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February 10th, 2013 at 10:18:17 PM permalink
Quote: stargazer

Crystalmath is correct. The area of the outer circle minus the area of the inner circle will always be equal to the area of a circle with a radius of 10'



I don't think so, because we do not know the diamater (or the radius) of the inner circle that the circular hallway goes around. (Imagine trying to carpet a circular racetrack around a circular field.) Since we don't know the length of the track, nor the diameter of the inner field, we cannot determine the area of the actual track itself. All we know is the width of the track, which isn't enough to compute the area of it.

I hope that I am not missing something...

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