Dimensions of a rectangle

Question

A rectangle has an area that is numerically twice its perimeter. If the length is twice the width, what are the dimensions?

Answer 1

Let width = x

Let length = 2x

 

Fundamentally, the perimeter and area would be:

 

Perimeter = 2(x) + 2(2x)

               = 2x + 4x

               = 6x

 

Area = 2x²

 

 

 

Area is also twice the perimeter, so

 

Area = 2(6x)

Area = 12x

 

 

Now we have two types of areas (in bold font):   The fundamental area, and the area related to the perimeter.  We can equate these areas, they are part of the same rectangle.

 

Fundamental area = Area related to perimeter

 

2x² = 12x

 

2x² - 12x = 0

 

2x(x - 6) = 0

 

x = 0     and        x = 6

 

 

We cannot have a zero dimension, so we accept  x=6

 

Substitute this value of x into the variables.

 

width = 6

length = 12

 

 

Now let's check.

 

Perimeter = 2(6 + 12)

               = 2(18)

               = 36

 

Area = 6 * 12

        = 72

 

 

Now, is the area twice the perimeter?

 

72 / 36 = 2

 

 

Yes it is.  This checks out.

 

 

Answer 2

Let the length and width be a and b, respectively. So its perimeter is 2a + 2b and area is ab.

 

According to the statement, ab = 2(2a + 2b) = 4a + 4b and a = 2b

 

Plugging this into the both sides of equation, ab = 4a + 4b --> 2b² = 8b + 4b = 12b

 

Dividing both the sides by 2b, b = 6 and a = 12

 

Dimensions of  the rectangle are 12 and 6 units.

Answer 3

Area of rectangle = Length (l) * Width  (w)

Perimeter of rectangle = 2 (l+w)

 

according to question, 

Area = 2 Perimeter

l*w  =  2* [2(l+w)]    -----  (1)

 

Now, given also,  l = 2w  ---- (2)        so if we replace this in the above equation --- (1), it becomes

 

(2w)*w = 2*[2(2w +w)]

2w^2 = 2*[2(3w)]

2w^2 = 12*w

2w^2 - 12*w = 0

2w (w - 6) = 0

 

Either 2w = 0

  i .e. w = 0

   so, equation -- (2) becomes,   l = 0   [Ignore this  set of answers ]

 

OR, w -6 = 0

i. e w = 6

 so, equation -- (2) becomes,    L= 2*6 = 12

 

So the answer is, L= 12, W = 6