Question 1196090
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The difference between the length and width of the room is 30-19=11 feet.  Since the strip around the rug is to be of uniform width, the difference between the length and width of the rug must also be 11 feet.<br>
So find how to factor 476 as the product of two integers whose difference is 11.<br>
476 = 119*4 = 7*17*4 = 28*17<br>
ANSWER: The rug should be 28 feet long and 17 feet wide.<br>
For a formal algebraic solution, we can let x be the width of the uniform strip around the rug, making the dimensions of the rug 30-2x and 19-2x.  Then, given the area of the rug as 476 square feet,<br>
{{{(30-2x)(19-2x)=476}}}
{{{570-98x+4x^2=476}}}
{{{4x^2-98x+94=0}}}
{{{2x^2-49x+47=0}}}
{{{(2x-47)(x-1)=0}}}<br>
The width of the uniform strip is either 47/2 = 23.5 feet or 1 foot.<br>
The 23.5 feet makes no sense, so the width of the strip is 1 foot, making the dimensions of the rug 28 feet by 17 feet.<br>
But that algebraic solution was much harder than the informal solution shown earlier -- so the formal algebra doesn't give you an easier way to find the answer to the problem.<br>