The area of a rectangle is 4 times the area of a square. The area of the square is 729 sq cm and the length of the rectangle is 81 cm. What is the difference between the side of the square and the breadth of the rectangle?

Options

A. 18 cm
B. 27 cm
C. 24 cm
D. 9 cm

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Find the side of the square
The area of the square is given as 729 sq cm. The formula for the area of a square is:

\[ \text{Area} = \text{side}^2 \]

To find the side of the square, we take the square root of the area:

\[ \text{side} = \sqrt{729} = 27 \text{ cm} \]

### Step 2: Calculate the area of the rectangle
According to the problem, the area of the rectangle is 4 times the area of the square. Therefore:

\[ \text{Area of rectangle} = 4 \times \text{Area of square} = 4 \times 729 = 2916 \text{ sq cm} \]

### Step 3: Use the length of the rectangle to find the breadth
The area of a rectangle can also be calculated using the formula:

\[ \text{Area} = \text{length} \times \text{breadth} \]

We know the length of the rectangle is 81 cm. Let \( B \) be the breadth of the rectangle. Thus, we can set up the equation:

\[ 81 \times B = 2916 \]

To find \( B \), we can rearrange the equation:

\[ B = \frac{2916}{81} \]

Calculating this gives:

\[ B = 36 \text{ cm} \]

### Step 4: Find the difference between the side of the square and the breadth of the rectangle
Now we need to find the difference between the side of the square and the breadth of the rectangle:

\[ \text{Difference} = \text{side of square} - \text{breadth of rectangle} = 27 \text{ cm} - 36 \text{ cm} \]

Calculating this gives:

\[ \text{Difference} = 27 - 36 = -9 \text{ cm} \]

Since we are looking for the absolute difference, we take the positive value:

\[ \text{Absolute Difference} = 9 \text{ cm} \]

### Final Answer
The difference between the side of the square and the breadth of the rectangle is **9 cm**.

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