Question

The area of a square field is 5184 \[{{\rm{m}}^2}\]. Find the area of a rectangular field, whose perimeter is equal to the perimeter of the square field, and whose length is twice of its breadth.

Answer 1

Hint: Here, we need to find the area of the rectangular field. First, we will find the side of the square and its perimeter. Using the formula for the perimeter of a rectangle, we can find the dimensions of the rectangle. Finally, we will substitute the value of dimensions of the rectangle in the formula for the area of a rectangle to find the area of the rectangular field.

Formula Used:
We will use the following formulas:
1.The area of a square is given by the formula \[{s^2}\], where \[s\] is the length of the side of the square.
2.We will use the formula Area of rectangle \[ = l \times b\], where \[l\] is the length of the rectangle and \[b\] is the breadth of the rectangle.

Complete step-by-step answer:
First, we will find the side of the square.
The area of a square is given by the formula \[{s^2}\], where \[s\] is the length of the side of the square.
It is given that the area of a square field is 5184 \[{{\rm{m}}^2}\].
Therefore, we get
\[5184 = {s^2}\]
Taking the square root of both sides, we get
 \[ \Rightarrow s = 72\] m
Thus, the side of the square is of length 72 m.
Now, we will find the perimeter of the square.
Substituting \[s = 72\] m in the formula for perimeter of a square, we get
\[ \Rightarrow \] Perimeter \[ = 4 \times 72\]
Multiplying the terms, we get
\[ \Rightarrow \] Perimeter \[ = 288\] m
Let the breadth of the rectangular field be \[x\].
It is given that the length of the rectangular field is twice its breadth.
Therefore, we get
\[ \Rightarrow \] Length of rectangular field \[ = 2x\]
Substituting breadth as \[x\] and length as \[2x\] in the formula for perimeter of a rectangle, we get
\[ \Rightarrow \] Perimeter \[ = 2\left( {2x + x} \right)\]
Adding the like terms in the parentheses, we get
\[ \Rightarrow \] Perimeter \[ = 2\left( {3x} \right)\]
Multiplying the terms in the expression, we get
\[ \Rightarrow \] Perimeter \[ = 6x\]
It is given that the perimeter of the square and rectangular field is the same.
Therefore, we get the equation
\[ \Rightarrow \] \[6x = 288\]
This is a linear equation in one variable. We will solve this to find the value of \[x\].
Dividing both sides of the equation by 6, we get
\[ \Rightarrow \] \[x = 48\]
Therefore, the breadth of the rectangular field is 48 m.
Substituting \[x = 48\] m in the expression \[2x\], we get
\[ \Rightarrow \] Length of rectangular field \[ = 2 \times 48 = 96\] m
Finally, we will calculate the area of the rectangular field.
The area of a rectangle is the product of its length and breadth.
Therefore, we get
\[ \Rightarrow \] Area of rectangular field \[ = 96 \times 48\]
Multiplying the terms in the expression, we get
 \[ \Rightarrow \] Area of rectangular field \[ = 4608{\rm{ }}{{\rm{m}}^2}\]
Thus, we get the area of the rectangular field as 4608 \[{{\rm{m}}^2}\].

Note: We have formed a linear equation in one variable in terms of \[x\] in the solution. A linear equation in one variable is an equation of the form \[ax + b = 0\], where \[a\] and \[b\] are integers. A linear equation of the form \[ax + b = 0\] has only one solution.
We used the formula for the perimeter of a square and rectangle in the solution.
The perimeter of a square of side \[s\] is given as \[4s\].
The perimeter of a rectangle with length \[l\] and breadth \[b\] is given as \[2\left( {l + b} \right)\].