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