Question

The difference between the length and breadth of a rectangle is 6 cm. The length of the rectangle is equal to the side of the square whose area is 729 sq. cm. What is the perimeter of the rectangle?
$
  \left( a \right)96 \\
  \left( b \right)108 \\
  \left( c \right)92 \\
  \left( d \right)88 \\
  \left( e \right){\text{None of these}} \\
 $

Answer 1

Hint-In this question, we use the concept of rectangle and square. We have to use area of square $ = {a^2}$ , where a is the side of square and perimeter of rectangle $ = 2\left( {l + b} \right)$ , where $l{\text{ and }}b$ is length and breadth of rectangle.

Complete step-by-step solution -
Consider a rectangle of length $l$ and breadth $b$ .
Now the difference between the length and breadth of a rectangle is 6 cm.
$ \Rightarrow l - b = 6.........\left( 1 \right)$
Now, we have a square side and an area is 729 sq. cm.
$
   \Rightarrow {\text{Area of square}} = {a^2} \\
   \Rightarrow {a^2} = 729 \\
$
Taking square root on both sides of equation,
$
   \Rightarrow a = \sqrt {729} \\
   \Rightarrow a = \pm 27 \\
$
We know length cannot be negative so we eliminate the negative value of a.
So, the value of a is 27cm.
Now, in question, given that the length of the rectangle is equal to the side of the square.
$
   \Rightarrow {\text{Length of rectangle}} = {\text{side of square}} \\
   \Rightarrow l = a \\
   \Rightarrow l = 27cm \\
 $
Now, put the value of $l$ in (1) equation.
$
   \Rightarrow 27 - b = 6 \\
   \Rightarrow b = 27 - 6 \\
   \Rightarrow b = 21cm \\
 $
Perimeter of rectangle $ = 2\left( {l + b} \right)$
$
   \Rightarrow 2\left( {27 + 21} \right) \\
   \Rightarrow 2 \times 48 \\
   \Rightarrow 96cm \\
$
So, the correct option is (a).

Note-In such types of problems we have to find the value of the side of the square by using the formula of area of square and then put the side of square equal to the length of the rectangle then find the value of breadth by using the relation given in the question. So after using the formula of the perimeter of the rectangle we will get the required answer.