The perimeter of a rectangle is 160 metres and the difference of two sides is 48 metres. Find the side of a square whose area is equal to the area of this rectangle?

Options

A. 32m
B. 8 m
C. 4m
D. 16 m

Answer

To solve the problem step by step, we will follow a systematic approach:

### Step 1: Understand the given information
We know that:
- The perimeter of the rectangle is 160 meters.
- The difference between the two sides (length and breadth) is 48 meters.

### Step 2: Set up the equations
1. The formula for the perimeter of a rectangle is:
\[ P = 2(l + b) \]
Given that \( P = 160 \), we can write:
\[ 2(l + b) = 160 \]
Dividing both sides by 2:
\[ l + b = 80 \quad \text{(Equation 1)} \]

2. The difference between the length and breadth is given as:
\[ l - b = 48 \quad \text{(Equation 2)} \]

### Step 3: Solve the equations
Now we will solve these two equations simultaneously.

**Adding Equation 1 and Equation 2:**
\[ (l + b) + (l - b) = 80 + 48 \]
This simplifies to:
\[ 2l = 128 \]
Dividing both sides by 2:
\[ l = 64 \quad \text{(Length)} \]

**Substituting \( l \) back into Equation 1 to find \( b \):**
\[ 64 + b = 80 \]
Subtracting 64 from both sides:
\[ b = 16 \quad \text{(Breadth)} \]

### Step 4: Calculate the area of the rectangle
The area \( A \) of the rectangle is given by:
\[ A = l \times b \]
Substituting the values of \( l \) and \( b \):
\[ A = 64 \times 16 = 1024 \, \text{square meters} \]

### Step 5: Find the side of the square with the same area
Let the side of the square be \( s \). The area of the square is given by:
\[ s^2 = A \]
Substituting the area we found:
\[ s^2 = 1024 \]
Taking the square root of both sides:
\[ s = \sqrt{1024} = 32 \, \text{meters} \]

### Final Answer
The side of the square whose area is equal to the area of the rectangle is **32 meters**.

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