SOLUTION: I would sincerely appreciate assistance with the following question: A rug is to cover two-thirds of the floor area of an 8 m X 12 m room. The uncovered part of the floor is to

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Question 465766: I would sincerely appreciate assistance with the following question:
A rug is to cover two-thirds of the floor area of an 8 m X 12 m room. The uncovered part of the floor is to form a strip of uniform width around the rug. Find the width of the strip.
Answer by kingme18(98) About Me  (Show Source):
You can put this solution on YOUR website!
I'm going to work with the rug first, finding the dimensions and the area. Based on that, I'll be able to solve for x (uniform width).

Rug's dimensions: If you draw a rug in the middle of a floor, you'll have bare floor on all sides. That means that if you consider the floor's width of 8 m, there is bare floor on EACH side of the rug--that's 2x, not just x. The width of the rug would be the total width minus 2x: 8-2x. Similarly, the length is 12-2x.

Rug's area: The total floor area is 8%2A12=96m%5E2. The rug's area is 2/3 of that, or %282%2F3%29%2A96=64m%5E2.

Length * Width = Area: %2812-2x%29%288-2x%29=64. Simplify the left side (you may know this as FOIL) to get 96+-+24x+-+16x+%2B+4x%5E2+=+64. Simplify more and put it in a prettier order: 4x%5E2+-+40x+%2B+96+=+64. We tend to like quadratics to equal 0, so subtract 64 on both sides: 4x%5E2+-+40x+%2B+32+=+0. Before moving on, I'd like to divide through by 4 (it's the GCF, and as long as you divide EVERYTHING by 4, it's totally legal): x%5E2+-+10x+%2B+8+=+0.

Unfortunately, this does not factor. Fortunately, we have the quadratic formula! x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ Here, a = 1, b = -10, and c = 8. x+=+%2810+%2B-+sqrt%28+%28-10%29%5E2-4%2A1%2A8%29%29%2F%282%2A1%29+
--> x+=+%2810+%2B-+sqrt%28100-32%29%29%2F2
--> x+=+%2810+%2B-+sqrt%2868%29%29%2F2
I don't like rounding, but at this point I'm going to round and say sqrt%2868%29=8.246
--> x+=+%2810+%2B-+8.246%29%2F2
Split that into two separate equations to finish simplifying:
x+=+%2810+%2B+8.246%29%2F2+=+18.246%2F2+=+9.123 --> x = 9.123
x+=+%2810+-+8.246%29%2F2+=+1.754%2F2+=+0.877 --> x = 0.877

The first answer, x = 9.123, makes no sense, since that means the bare floor would be longer than the width of the floor (8 m)...that answer is wrong.

Hopefully the other answer, x = 0.877, is right. To check, I'm going to find the dimensions of the rug: 8+-+2%2A0.877+=+6.246 and 12+-+2%2A0.877+=+10.246. Finally, 6.246%2A10.246=63.9965, which (considering that I rounded) is pretty close to 64 :)