The perimeter of a square is double than the perimeter of a rectangle. The area of the rectangle is 36 sq.cm. what is the area of square?

Options

A. 72 sq.cm
B. 56 sq.cm
C. 64 sq.cm
D. can't be determined

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the relationship between the perimeters
We know that the perimeter of a square is double the perimeter of a rectangle.

Let the perimeter of the rectangle be \( P_r \). Then, the perimeter of the square \( P_s \) can be expressed as:
\[ P_s = 2 \times P_r \]

### Step 2: Write the formulas for the perimeters
The perimeter of a rectangle is given by:
\[ P_r = 2 \times (l + b) \]
where \( l \) is the length and \( b \) is the breadth of the rectangle.

The perimeter of a square is given by:
\[ P_s = 4 \times s \]
where \( s \) is the side of the square.

### Step 3: Set up the equation
From the relationship established in Step 1, we can equate the two expressions for the perimeter:
\[ 4s = 2 \times (2(l + b)) \]
This simplifies to:
\[ 4s = 4(l + b) \]
Dividing both sides by 4 gives us:
\[ s = l + b \]

### Step 4: Use the area of the rectangle
We are given that the area of the rectangle is \( 36 \, \text{sq.cm} \):
\[ l \times b = 36 \]

### Step 5: Solve for the side of the square
From the equation \( s = l + b \), we can express \( b \) in terms of \( l \):
\[ b = \frac{36}{l} \]
Substituting this into the equation for \( s \):
\[ s = l + \frac{36}{l} \]

### Step 6: Find the minimum value of \( s \)
To find the minimum value of \( s \), we can use calculus or the AM-GM inequality. However, we can also find integer pairs \( (l, b) \) that satisfy \( l \times b = 36 \):
- \( (1, 36) \)
- \( (2, 18) \)
- \( (3, 12) \)
- \( (4, 9) \)
- \( (6, 6) \)

Calculating \( s \) for these pairs:
1. For \( (1, 36) \): \( s = 1 + 36 = 37 \)
2. For \( (2, 18) \): \( s = 2 + 18 = 20 \)
3. For \( (3, 12) \): \( s = 3 + 12 = 15 \)
4. For \( (4, 9) \): \( s = 4 + 9 = 13 \)
5. For \( (6, 6) \): \( s = 6 + 6 = 12 \)

The minimum value of \( s \) is \( 12 \).

### Step 7: Calculate the area of the square
The area of the square is given by:
\[ \text{Area of square} = s^2 = 12^2 = 144 \, \text{sq.cm} \]

### Final Answer
The area of the square is \( 144 \, \text{sq.cm} \).

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