Hello, Kstrikeoutgirl!
Originally the dimensions of a rectangle were 23 cm by 20 cm.
When both dimensions were decreased by the same amount,
the area of the rectangle decreased by 120 cm².
Find the dimensions of the new rectangle.
We already know the following:
. . The original length was 23 cm.
. . The original width was 20 cm.
. . The original area was: \(\displaystyle 23\,\times\,20\:=\;460\) cm²
Let x be the amount of decrease in the length and width.
. . The new length is: \(\displaystyle 23\,-\,x\) cm.
. . The new width is: \(\displaystyle 20\,-\,x\) cm.
. . The new area is: \(\displaystyle (23\,-\,x)(20\,-\,x)\) cm².
The new area is 120 cm² less than the original area: \(\displaystyle 460\,-\,120\:=\:340\) cm².
There is our equation: \(\displaystyle \;(23\,-\,x)(20\,-\,x)\:=\:340\)
. . which simplifies to: \(\displaystyle \;x^2\,-\,43x\,+\,120\:=\:0\)
. . and factors: \(\displaystyle \;(x\,-\,3)(x\,-\,40)\:=\:0\)
. . and has roots: \(\displaystyle \;x\,=\,3\) and \(\displaystyle 40\)
Since the original rectangle was 23-by-20, we can't reduce the dimensions by 40 cm,
. . so: \(\displaystyle \;x\,=\,3\) cm
Don't forget to answer the question (the dimensions).
The new length is: \(\displaystyle \,23\,-\,3\:=\:20\) cm.
The new width is: \(\displaystyle \;20\,-\,3\:=\:17\) cm.
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<u>Check</u>
The new rectangle has an area of: \(\displaystyle \;20\,\times\,17\:=\:340\) cm².
This
is 120 cm² less than the original area (460) . . .
check!
