document.write( "Question 110528: a rectangle is three times as long as it is wide. if its length and width are both decreased by 2 cm, then its area is decreased by 36 cm^2. find its original dimensions. \n" ); document.write( "
Algebra.Com's Answer #80839 by bucky(2189)\"\" \"About 
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Let's begin with the original rectangle. If we define its width as X, then the length (which
\n" ); document.write( "we are told was three times as long as the width) is 3*X. The area of this rectangle is
\n" ); document.write( "found by multiplying the width times the length, so the area is X times 3X and this is
\n" ); document.write( "3X^2.
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\n" ); document.write( "Now suppose that each side is shortened by 2 cm. That would make the new width X - 2 cm and
\n" ); document.write( "the new length 3X - 2. The area of this new rectangle is the product of (X - 2) times (3X - 2).
\n" ); document.write( "Multiplying these two sides results in X*3X - 2X - 6X + 4. Simplifying this results in
\n" ); document.write( "the area of this new rectangle being 3X^2 - 8X + 4.
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\n" ); document.write( "The problem tells you that the difference in these two areas as being 36 cm^2. The difference
\n" ); document.write( "in the two areas is:
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\n" ); document.write( "3X^2 - (3X^2 - 8X + 4)
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\n" ); document.write( "The parentheses are preceded by a minus sign. You can remove the parentheses and the minus
\n" ); document.write( "sign if you change the signs of each of the terms inside the parentheses. When you do
\n" ); document.write( "this the difference in the areas of the two rectangles is defined by the expression:
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\n" ); document.write( "3X^2 - 3X^2 + 8X - 4
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\n" ); document.write( "The 3X^2 and - 3X^2 cancel each other out and you are left with:
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\n" ); document.write( "8X - 4
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\n" ); document.write( "as the difference in the areas of the two rectangles. But the problem tells you that this
\n" ); document.write( "difference is 36 cm^2. So you can write the equation:
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\n" ); document.write( "8X - 4 = 36
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\n" ); document.write( "to solve this, get rid of the -4 on the left side by adding +4 to both sides to get:
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\n" ); document.write( "8X = 40
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\n" ); document.write( "Finally solve for X by dividing both sides by 8 to get:
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\n" ); document.write( "X = 40/8 = 5
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\n" ); document.write( "Since X was defined as the width of the original rectangle and its length was 3X you know
\n" ); document.write( "that the dimensions of the original rectangle was 5 cm by 15 cm which gives an area of
\n" ); document.write( "5 times 15 = 75 cm^2
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\n" ); document.write( "The new rectangle has a length and width that are each 2 cm less. So the new rectangle has
\n" ); document.write( "a width of 5 - 2 or 3 cm. And the length of the new rectangle is 15 - 2 or 13 cm. This
\n" ); document.write( "means that the area of this new rectangle is 3 times 13 = 39 cm^2
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\n" ); document.write( "Check ... the difference in the areas of these two rectangles is 75 - 39 = 36 cm^2. This
\n" ); document.write( "is just as the problem said it should be.
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\n" ); document.write( "Hope this helps you to see your way through the problem and to see how you can work it
\n" ); document.write( "through to a solution.
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