The length of a rectangle is 3 times its width. If the width is increased by 2 cm and the length is increased by 4 cm, its area will be doubled. Find the origin

Question

The length of a rectangle is 3 times its width. If the width is increased by 2 cm and the length is increased by 4 cm, its area will be doubled. Find the original

Answer 1

Let:

    x = original length of rectangle

    y = original width of rectangle

 

We know that:

  original area = xy

 

Translate:

  "length of a rectangle is 3 times its width"     means

      x                          =  3   *     y                 [eq1]

 

"width is increased by 2 cm and the length is increased by 4 cm, its area will be doubled"   means

         ( y      +         2)         *        (  x                    +    4   )    =           2 * xy       [eq2]

 

The math:

   Use eq1 to substitute for x in eq2:

        ( y + 2 ) * ( 3y + 4 ) = 2 * (3y)(y)

          3y² + 10y + 8 =  6y²             [collect terms and use F-O-I-L]

         3y² - 10y - 8 = 0                    [get 0 on right side]

          (y - 4)(3y + 2) = 0              [factor or use Quadratic Formula]

    Either y=4 or y=-2/3                 [multiplicative property of zero]

  y cannot be -2/3 because it is the width of the rectangle, so y=4  [original width]

    Using eq1, x=12    [original length]

 

Check:

  Is    12 = 3*4   ?

         12 = 12   ?yes

  Is    (12+4)(4+2) = 2(12)(4)

           (16)(6) = 2(12)(4)

             96 = 96   ?yes