The area of a square field is $5184\ m^2$. A rectangular field, whose length is twice its breadth has its perimeter equal to the perimeter of the square field. Find the area of the rectangular field.

Given:

The area of a square field is $5184\ m^2$. A rectangular field, whose length is twice its breadth has its perimeter equal to the perimeter of the square field.

To do:

We have to find the area of the rectangular field.

Solution:

Let the side of the square be $x$.

Therefore,

$x\times x=5184$

$x^{2}=5184$

$x=\sqrt{5184}$      (Prime factorisation of 5184 is $2 \times 2 \times 2 \times 2 \times 2 \times2\times3 \times 3 \times 3 \times 3$)

$=2 \times 2 \times 2 \times 3 \times 3$

$=72$

The side of the square is $72\ m$.

 Perimeter of the square field $= 72 \times 4\ m$

$= 288\ m$

This implies,

Perimeter of the rectangle $=288\ m$

Let the breadth of the rectangular field be $x$

This implies,

Length of the rectangular field $= 2x$

Perimeter $= 2 (l + b)$

$= 2 (2x + x)$

$= 2\times 3x$

$= 6x$

$6x=288$

$x=\frac{288}{6}$

$x=48\ m$

Length of the rectangular field$= 2x = 2 \times 48$

$= 96\ m$

Breadth $=x = 48\ m$

Area of the rectangular field $= 96 \times 48\ m^2$

$= 4608\ m^2$