Perimeter of a rectangle and a square are equal, Perimeter of the square is 96 cm and breadth of the rectangle is 8 cm less than the side of the square. Then, two times the area (in `cm^(2)` ) of the rectangle is

Options

A. 1024
B. 1086
C. 1087
D. 1089

Answer

To solve the problem step by step, we will follow the given information and use the formulas for perimeter and area.

### Step 1: Find the side length of the square
The perimeter of a square is given by the formula:
\[ \text{Perimeter} = 4 \times \text{side} \]
Given that the perimeter of the square is 96 cm, we can set up the equation:
\[ 4 \times \text{side} = 96 \]
To find the side length, we divide both sides by 4:
\[ \text{side} = \frac{96}{4} = 24 \text{ cm} \]

### Step 2: Determine the breadth of the rectangle
According to the problem, the breadth of the rectangle is 8 cm less than the side of the square. Therefore, we can calculate the breadth as follows:
\[ \text{breadth} = \text{side} - 8 = 24 - 8 = 16 \text{ cm} \]

### Step 3: Find the perimeter of the rectangle
Since the perimeter of the rectangle is equal to the perimeter of the square, we have:
\[ \text{Perimeter of rectangle} = 96 \text{ cm} \]
The perimeter of a rectangle is given by the formula:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{breadth}) \]
Let the length of the rectangle be \( l \). We can set up the equation:
\[ 2 \times (l + 16) = 96 \]
Dividing both sides by 2 gives:
\[ l + 16 = 48 \]
Now, subtracting 16 from both sides gives:
\[ l = 48 - 16 = 32 \text{ cm} \]

### Step 4: Calculate the area of the rectangle
The area \( A \) of a rectangle is calculated using the formula:
\[ A = \text{length} \times \text{breadth} \]
Substituting the values we found:
\[ A = 32 \times 16 = 512 \text{ cm}^2 \]

### Step 5: Find two times the area of the rectangle
To find two times the area, we simply multiply the area by 2:
\[ 2 \times A = 2 \times 512 = 1024 \text{ cm}^2 \]

### Final Answer
Thus, two times the area of the rectangle is:
\[ \boxed{1024 \text{ cm}^2} \]