Question

The length of a rectangle is 4 m longer than its width. if the perimeter of the rectangle is 36 m , find its area.

Answer 1

Answer:   The area of the rectangle is:  " 77 m² " .

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Note:  The formula for the area, "A" of a rectangle:

 →  A  =  L  *  w   ;

                 in which:  
     A = "area (of rectangle)" ; [in units of "m² " ; that is:  "square meters" ] ; 
                                     
     L = length = "(4 + w)" {in units of "meters (m)" } ;  
             
     w = width  {in units of "meters (m)" } ; 
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So;  " A = L * w " ;  

Substitute the known expression for the "length, L" ;  & rewrite the formula for the given area of OUR area for the rectangle in OUR GIVEN PROBLEM:

              →  A = (4 + w) * w '' ;
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Note the formula for the perimeter, "P" ; 

 →  P = 2L + 2w  ; 

↔   2L + 2w = P 

 →  2L + 2w  = 36 m ;
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We want to find the "area" , "A" :
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Using the formula for the "perimeter, "P" (of the rectangle) ; & given that the perimeter is:  "36" (meters) ;  

 →  2L + 2w  = 36 ;

 →  Let us plug in the values for "Length (L)" & "width (w)" ; 

 →  2(w + 4)  + 2w = 36  ; 

So;   (2*w) + (2*4) + 2w = 36 ;  Solve for "w" ;

    →  2w + 8 + 2w = 36 ; 

    → Combine the "like terms" :

         + 2w + 2w = 4w ; 

   →  And rewrite: 

         4w + 8 = 36 ; 

Now, subtract "8" from EACH SIDE of the equation:

         4w + 8 − 8 = 36 − 8 ; 

to get: 

         4w = 28 ; 

Now, divide EACH SIDE of the equation by "4" ; 
      to isolate "w" on EACH SIDE of the equation ; & to solve for "w" ; 

         4w / 4 = 28 / 4  ; 

           → w = 7 ;   → The "width" of the rectangle is:  " 7 m " .

Now, we can find the "length" of the rectangle:

The length, "L" , of the rectangle = 4 + w = 4 + 7 = 11 .
      
           →  L = 11 .  →  The "length" of the rectangle is:  " 11 m " .
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Now, we can find the area, "A", of the rectangle.

 A = L * w  =  11 m * 7 m  =  " 77 m² " .

  →  The area of the rectangle is:  " 77 m² " .
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To check our answer:
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→  " P = 2L + 2w "  ; 

Given that "P = 36 m" ; 

Plug in "36 m" (for "P") ; into the equation ;

and plug in our calculated values for
                        "length, L" (which is "11 m") ; & "width, w" (which is "7 m") ; 

to see if the equation holds true ; that is, to see if both sides of the equation are equal ;
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 →  36 m = ?  2L + 2w ?? ;

 →  36 m = ?  2(11 m) + 2(7 m) ?? ; 

 →  36 m = ?  22 m + 14 m ?? ; 

 →  36 m = ?  36 m ?  Yes! 
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