Hello again, polakio92!
c) A rectangle is cut to form a square.
The perimeter of the rectangle is 54 cm, and the area of the cut-off piece is 36 cm².
What are the dimensions of the original rectangle? Code:
x y
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x | | | x
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x y
The perimeter of the rectangle is 54: .$\displaystyle 4x + 2y \:=\:54\quad\Rightarrow\quad y \:=\:27 - 2x\;\;{\color{blue}[1]}$
The area of the cut-off piece is 36 cm²: .$\displaystyle xy \:=\:36\;\;{\color{blue}[2]}$
Substitute [1] into [2]: .$\displaystyle x(27-2x) \:=\:36\quad\Rightarrow\quad 2x^2 - 27x + 36 \:=\:0$
. . which factors: .$\displaystyle (x-12)(2x-3)\:=\:0$
. . and has roots: .$\displaystyle x \;=\;12,\:\frac{3}{2}$
and has y-values: .$\displaystyle y \;=\;3,\:24$
There are two solutions: .$\displaystyle (x,y) \;=\;(12,\,3)\text{ and }(1.5,\:24)$
The original rectangle was: . $\displaystyle \boxed{12 \times 15}\;\text{ or }\;\boxed{1.5 \times 25.5}$
Code:
12 3
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12 | | | 12
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12 3
1½ 24
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1½ | | | 1½
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1½ 24
Both of them satisfy the problem!