The area of a rectangle is `216m^2` with the respective ratio between its length and breadth as `3:2` The diameter of a circle is `1'1/6` the of the breadth of the rectangle. What is the difference between the area of the rectangle and that of the circle? `(in m^2)`

Options

A. 62
B. 66
C. 70
D. 68

Answer

To solve the problem step by step, we will follow the given information about the rectangle and the circle.

### Step 1: Determine the dimensions of the rectangle
The area of the rectangle is given as \( 216 \, m^2 \). The ratio of the length (L) to the breadth (B) is \( 3:2 \).

Let:
- Length \( L = 3x \)
- Breadth \( B = 2x \)

Using the area formula for a rectangle:
\[ L \times B = 216 \]
Substituting the expressions for L and B:
\[ (3x) \times (2x) = 216 \]
\[ 6x^2 = 216 \]
\[ x^2 = \frac{216}{6} = 36 \]
\[ x = 6 \]

Now substituting back to find L and B:
\[ L = 3x = 3 \times 6 = 18 \, m \]
\[ B = 2x = 2 \times 6 = 12 \, m \]

### Step 2: Calculate the diameter of the circle
The diameter of the circle is given as \( 1 \frac{1}{6} \) times the breadth of the rectangle.

First, convert \( 1 \frac{1}{6} \) to an improper fraction:
\[ 1 \frac{1}{6} = \frac{7}{6} \]

Now, calculate the diameter:
\[ \text{Diameter} = \frac{7}{6} \times B = \frac{7}{6} \times 12 = 14 \, m \]

### Step 3: Calculate the radius of the circle
The radius \( r \) of the circle is half of the diameter:
\[ r = \frac{14}{2} = 7 \, m \]

### Step 4: Calculate the area of the circle
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Substituting the radius:
\[ A = \pi \times (7)^2 = 49\pi \, m^2 \]

### Step 5: Calculate the difference between the area of the rectangle and the area of the circle
Now, we need to find the difference:
\[ \text{Difference} = \text{Area of Rectangle} - \text{Area of Circle} \]
Substituting the values:
\[ \text{Difference} = 216 - 49\pi \]

To get a numerical approximation, use \( \pi \approx 3.14 \):
\[ 49\pi \approx 49 \times 3.14 \approx 153.86 \]
Thus,
\[ \text{Difference} \approx 216 - 153.86 \approx 62.14 \, m^2 \]

### Final Answer
The difference between the area of the rectangle and that of the circle is approximately \( 62.14 \, m^2 \).

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