A circle circumscribes a rectangle whose sides are in the ratio of 4 :3 If the area of the rectangle is `192 cm^(2)`, then the perimeter (in cm) of the circle is :

Options

A. `20 pi`
B. `10 pi`
C. `12 pi`
D. `15pi`

Answer

To solve the problem step by step, let's follow the given information and apply the necessary mathematical concepts.

### Step 1: Understand the problem
We have a rectangle whose sides are in the ratio of 4:3, and the area of the rectangle is given as 192 cm². We need to find the perimeter of the circle that circumscribes this rectangle.

### Step 2: Set up the dimensions of the rectangle
Let the length of the rectangle be \( 4x \) and the breadth be \( 3x \), where \( x \) is a common multiplier.

### Step 3: Write the equation for the area of the rectangle
The area \( A \) of the rectangle can be expressed as:
\[ A = \text{Length} \times \text{Breadth} = (4x) \times (3x) = 12x^2 \]
Given that the area is 192 cm², we can set up the equation:
\[ 12x^2 = 192 \]

### Step 4: Solve for \( x^2 \)
To find \( x^2 \), divide both sides of the equation by 12:
\[ x^2 = \frac{192}{12} = 16 \]

### Step 5: Find \( x \)
Taking the square root of both sides gives:
\[ x = \sqrt{16} = 4 \]

### Step 6: Calculate the dimensions of the rectangle
Now we can find the length and breadth:
- Length = \( 4x = 4 \times 4 = 16 \) cm
- Breadth = \( 3x = 3 \times 4 = 12 \) cm

### Step 7: Calculate the diameter of the circumscribing circle
The diameter \( D \) of the circle can be calculated using the Pythagorean theorem:
\[ D = \sqrt{(\text{Length})^2 + (\text{Breadth})^2} = \sqrt{(16)^2 + (12)^2} \]
Calculating this gives:
\[ D = \sqrt{256 + 144} = \sqrt{400} = 20 \text{ cm} \]

### Step 8: Find the radius of the circle
The radius \( r \) is half of the diameter:
\[ r = \frac{D}{2} = \frac{20}{2} = 10 \text{ cm} \]

### Step 9: Calculate the perimeter (circumference) of the circle
The perimeter \( P \) of the circle is given by the formula:
\[ P = 2\pi r \]
Substituting the value of the radius:
\[ P = 2 \times \pi \times 10 = 20\pi \text{ cm} \]

### Final Answer
The perimeter of the circle is \( 20\pi \) cm.

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