Question

The length of a rectangle is twice its width. Find its area, if its perimeter is 7 1/3 cm.

Answer 1

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 $\frac{1}{3}$ cm" ; Solve for "A" ;
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7 $\frac{1}{3}$ cm = 2L + 2w ; 

Divide EACH side by "2" ;
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  7 $\frac{1}{3}$ cm / 2  = (2L + 2w) / 2 ; 

to get:  

  7 $\frac{1}{3}$ cm / 2 = L + w ;

Given:  L = 2w ;  rewrite the above equation; substituting "2w" for "L" ;
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  7 $\frac{1}{3}$ cm / 2 = 2w + w ;


 7 $\frac{1}{3}$ cm / 2 = 3w ;
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Note:  
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7 $\frac{1}{3}$ cm / 2 ;

  =  $\frac{[(3*7) + 1]}{3}$ cm / 2 ;

  =  $\frac{22}{3}$ cm / 2 ;

  = $\frac{22 cm}{3}$  * $\frac{1}{2}$ ;

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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   $\frac{11 cm}{3}$  * $\frac{1}{1}$ ;

and further simplify:

 $\frac{11 cm}{3}$ * $\frac{1}{1}$ ;

   =  $\frac{11 cm}{3}$  * 1  ;

   =  $\frac{11 cm}{3}$  ;
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Now, we can take the equation:

 7 $\frac{1}{3}$ cm / 2 = 3w ;
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and rewrite as:
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$\frac{11 cm}{3}$   = 3w ; 

and multiply each side by "$\frac{1}{3}$" ; to isolate "w" on one side of the equation ; and to solve for "w" ; 

   $\frac{11 cm}{3}$  * $\frac{1}{3}$ = {3w} * $\frac{1}{3}$ ;

to get:  

    $\frac{11 cm * 1}{3*3}$ = w ;

          $\frac{11 cm}{9}$ =  w ;

↔  w = $\frac{11}{9}$ cm ;
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Given:  L = 2w ;  

L = 2 * $\frac{11}{9}$ cm ;

      L = { $\frac{2}{1}$ * $\frac{11}{9}$ } cm ; 

      L = $\frac{2*11}{1*9}$ cm ;

      L = $\frac{22}{9}$ cm ;
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Now, solve for "A" (area):

    A = L * w ; 

     A   = $\frac{22}{9}$ cm * $\frac{11}{9}$ cm ;

     A   =  $\frac{22* 11}{9*9}$ cm² ;

     A   =  $\frac{242}{81}$ cm² ;

     A = 2.9876543209876543 cm² ;  round to:  "2.989 cm² " .
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