document.write( "Question 1096124: A rectangular rug has a decorative interior with a 1/2 foot border of uniform width around the outside. The length of the decorative area is 3 feet more than the width. If the area of the rug (including the border) is 108 ft2, find the dimensions of the rug (including the border). Show all work. Answer with a complete sentence. \n" ); document.write( "

Algebra.Com's Answer #710665 by greenestamps(13200) $\"\"$ $\"About$

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If the length of the decorative area is 3 more than the width, and the border is uniform width, then the length of the whole rug will still be 3 more than the width. So the problem is easier to solve if you just solve the equation

\n" ); document.write( " $\"x%28x%2B3%29+=+108\"$
\n" ); document.write( " $\"x%5E2%2B3x+=+108\"$
\n" ); document.write( " $\"x%5E2%2B3x-108+=+0\"$
\n" ); document.write( " $\"%28x%2B12%29%28x-9%29+=+0\"$

\n" ); document.write( "x = -12 or x = 9; obviously the positive answer is the one that makes sense.

\n" ); document.write( "Since the problem asks for the area of the rug including the border, we know the width is 9 and the length is x+3 = 12.

\n" ); document.write( "Note that solving the problem algebraically involves solving a quadratic equation; to do that by factoring, we need to find two numbers whose difference is 3 and whose product is 108.

\n" ); document.write( "But that is exactly what the problem asks us to do, WITHOUT doing any algebra: find a length and a width whose product is 108, with the length 3 more than the width.

\n" ); document.write( "So using formal algebra to solve this problem is a waste of time -- except as practice in solving problems using algebra. \n" ); document.write( "

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