The area of a square is twice the area of a rectangle. If the area of the square is 324 sq. cms and the length of the rectangle is 13.5 cms, what is the difference between the bredth of the rectangle and the side of the square ?

Options

A. 8 cms
B. 10 cms
C. 12 cms
D. 6 cms

Answer

To solve the problem step by step, we will follow the information given in the question and perform the necessary calculations.

### Step 1: Understand the relationship between the areas
We know that the area of the square is twice the area of the rectangle.
Let:
- Area of the square = A_square
- Area of the rectangle = A_rectangle

From the problem, we have:
\[ A_{square} = 2 \times A_{rectangle} \]

### Step 2: Substitute the area of the square
We are given that the area of the square is 324 sq. cms. Therefore, we can substitute this value into the equation:
\[ 324 = 2 \times A_{rectangle} \]

### Step 3: Solve for the area of the rectangle
To find the area of the rectangle, we rearrange the equation:
\[ A_{rectangle} = \frac{324}{2} = 162 \text{ sq. cms} \]

### Step 4: Use the dimensions of the rectangle
We know the length of the rectangle (l) is 13.5 cms. The area of a rectangle is given by the formula:
\[ A_{rectangle} = l \times b \]
Where b is the breadth of the rectangle. We can substitute the known values:
\[ 162 = 13.5 \times b \]

### Step 5: Solve for the breadth of the rectangle
To find the breadth (b), we rearrange the equation:
\[ b = \frac{162}{13.5} \]

Now we perform the division:
\[ b = 12 \text{ cms} \]

### Step 6: Find the side of the square
The area of the square is given as 324 sq. cms. The side of the square (s) can be found using the formula:
\[ A_{square} = s^2 \]
So we have:
\[ s^2 = 324 \]

To find s, we take the square root:
\[ s = \sqrt{324} = 18 \text{ cms} \]

### Step 7: Calculate the difference between the breadth of the rectangle and the side of the square
Now we need to find the difference:
\[ \text{Difference} = s - b = 18 - 12 \]

### Step 8: Final calculation
Calculating the difference gives us:
\[ \text{Difference} = 6 \text{ cms} \]

Thus, the final answer is:
**The difference between the breadth of the rectangle and the side of the square is 6 cms.**

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