Answer: " 2.989 cm² " .
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Explanation:
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Let "A" represent the "area" ;
"L" represent the "Length" ;
"w" represent the "width" ;
"P" represent the "Perimeter".
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Note the following equations /formulas for a rectangle:
A = L * w ;
P = 2L + 2w ;
Given: "L = 2w " ;
and: " P = 7 [tex] \frac{1}{3}[/tex] cm" ; Solve for "A" ;
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7 [tex] \frac{1}{3}[/tex] cm = 2L + 2w ;
Divide EACH side by "2" ;
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7 [tex] \frac{1}{3}[/tex] cm / 2 = (2L + 2w) / 2 ;
to get:
7 [tex] \frac{1}{3}[/tex] cm / 2 = L + w ;
Given: L = 2w ; rewrite the above equation; substituting "2w" for "L" ;
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7 [tex] \frac{1}{3}[/tex] cm / 2 = 2w + w ;
7 [tex] \frac{1}{3}[/tex] cm / 2 = 3w ;
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Note:
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7 [tex] \frac{1}{3}[/tex] cm / 2 ;
= [tex] \frac{[(3*7) + 1]}{3}[/tex] cm / 2 ;
= [tex] \frac{22}{3}[/tex] cm / 2 ;
= [tex] \frac{22 cm}{3} [/tex] * [tex] \frac{1}{2} [/tex] ;
Note: The "22 cm" cancels to "11 cm" ; and the "2" cancels out to "1" ;
{Since: "(22 cm ÷ 2 = 11 cm)" ; and since: "(2 ÷ 2 = 1)" .
And we can rewrite the expression as:
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[tex] \frac{11 cm}{3} [/tex] * [tex] \frac{1}{1} [/tex] ;
and further simplify:
[tex] \frac{11 cm}{3} [/tex] * [tex] \frac{1}{1} [/tex] ;
= [tex] \frac{11 cm}{3} [/tex] * 1 ;
= [tex] \frac{11 cm}{3} [/tex] ;
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Now, we can take the equation:
7 [tex] \frac{1}{3}[/tex] cm / 2 = 3w ;
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and rewrite as:
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[tex] \frac{11 cm}{3} [/tex] = 3w ;
and multiply each side by "[tex] \frac{1}{3} [/tex]" ; to isolate "w" on one side of the equation ; and to solve for "w" ;
[tex] \frac{11 cm}{3} [/tex] * [tex] \frac{1}{3} [/tex] = {3w} * [tex] \frac{1}{3} [/tex] ;
to get:
[tex] \frac{11 cm * 1}{3*3} [/tex] = w ;
[tex] \frac{11 cm}{9} [/tex] = w ;
↔ w = [tex] \frac{11}{9} [/tex] cm ;
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Given: L = 2w ;
L = 2 * [tex] \frac{11}{9} [/tex] cm ;
L = { [tex] \frac{2}{1}[/tex] * [tex] \frac{11}{9} [/tex] } cm ;
L = [tex] \frac{2*11}{1*9} [/tex] cm ;
L = [tex] \frac{22}{9} [/tex] cm ;
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Now, solve for "A" (area):
A = L * w ;
A = [tex] \frac{22}{9} [/tex] cm * [tex] \frac{11}{9} [/tex] cm ;
A = [tex] \frac{22* 11}{9*9} [/tex] cm² ;
A = [tex] \frac{242}{81}
[/tex] cm² ;
A = 2.9876543209876543 cm² ; round to: "2.989 cm² " .
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