Question

The area of a square is 49 sq cm. A rectangle has the same perimeter as the square. If the length of the rectangle is 9.3 cm, What is its breadth? Also, find which has a greater area?

Answer 1

Hint: Here we will first let the breadth of the rectangle to be b. Now we will apply the formula of the area of the square to find its side and then apply the formula of the perimeter of the square and put it equal to the perimeter of the rectangle to find the breadth.
Now in order to find which has a greater area, we will find the area of rectangle using its formula and then compare the areas of both the figures.

Complete Step-by-step Solution
Let the breadth of the rectangle be $b$.

Now since it is given that the area of the square is 49 sq cm.
And we know that the area of the square is given by:
area \[= {a^2}\] where \[a\] is the length of side of square
Hence putting in the value we get:-
\[ {a^2} = 49 \]
$ a = \sqrt {49} $
$ a = 7cm $
Hence the length of the side of the square is 7cm
Now we will find the perimeter of the square.
Since we know that the perimeter of the square is given by:-
perimeter \[= 4a\], where \[a\] is the length of side of square
Therefore putting in the value of \[a\] we get:-
 perimeter \[ = 4\left( 7 \right) \]
  perimeter $= 28cm $
Now since it is given that the perimeters of the square and the rectangle are equal
And we know that the perimeter of the rectangle is given by:
perimeter \[= 2\left( {l + b} \right)\], where \[l\] is the length and \[b\] is the breadth of the rectangle.
Hence putting in the known values we get:-
$\Rightarrow 28 = 2\left( {9.3 + b} \right) $
$\Rightarrow 28 = 18.6 + 2b $
On simplification,
$\Rightarrow 2b = 28 - 18.6 $
$ \Rightarrow 2b = 9.4 $
On further simplification,
$\Rightarrow b = \dfrac{{9.4}}{2}$
$\Rightarrow b = 4.7cm $
Hence the breadth of the rectangle is \[4.7cm\]
Now we will calculate the area of the rectangle
Now as we know that the area of the rectangle is given by:-
area\[ = l \times b\], where \[l\] is the length and \[b\] is the breadth of the rectangle.
Putting in the known values we get:-
area\[ = 9.3 \times 4.7 \]
area \[= 43.71{\text{sq cm}} \]
Now the area of the square is 49 sq cm.
And the area of the rectangle is 43.71 sq cm.

$\therefore $ The square has a greater area in comparison to the rectangle. So, Option (D) is the correct option.

Note:
Students many times make mistakes while multiplying to matrices, they should always keep in mind that while multiplying two matrices we multiply respective elements moving from left to right in rows and from top to bottom in columns. Also, the diagonal entries should be written in a way that they go from top left to right bottom and not from top right to left bottom.