word problem

May 2010
3
0
if the length and the width of a rectangle are increased by two inches, the area of the rectangle is 120 square inches. if the length and the width of a rectangle are decreased by two inches, the area of the rectangle is 48 square inches. find the length of the rectangle.
 

Soroban

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Hello, mojojojo!

If the length and the width of a rectangle are increased by two inches,
the area of the rectangle is 120 square inches.
If the length and the width of a rectangle are decreased by two inches,
the area of the rectangle is 48 square inches.
Find the length of the rectangle.

Let \(\displaystyle L\) = length of the rectangle.
Let \(\displaystyle W\) = width of the rectangle.

Then we have: .\(\displaystyle \begin{array}{ccccc}(L+2)(W+2) &=& 120 & [1] \\
(L-2)(W-2) &=& 48 & [2]\end{array}\)

\(\displaystyle \begin{array}{cccccc}
\text{From [1] we have:} & LW + 2L + 2W + 4 &=& 120 \\
\text{From [2] we have:} & LW - 2L - 2W + 4 &=& 48 \end{array}\)

. . Subtract: . \(\displaystyle 4L + 4W \:=\:72 \quad\Rightarrow\quad W \:=\:18-L\)


Substitute into [1]: . \(\displaystyle (L+2)(20-L) \:=\:120 \quad\Rightarrow\quad L^2 - 18L + 80 \:=\:0\)

. . \(\displaystyle (L-10)(L-8) \:=\:0 \quad\Rightarrow\quad L \;=\;10,\:8\)


Assuming \(\displaystyle L > W\) we have: .\(\displaystyle L \:=\:10\text{ inches.}\)