The perimeter of a rectangle is x cm and circumference of a circle is 8 cm more than the perimeter of the rectangle. The ratio of radius of circle and. length of the rectangle is `1:2` and the ratio of length and breadth of rectangle is `7:3`. Find the length of the rectangle (in cm).

Options

A. 14
B. 21
C. 28
D. 35

Answer

To solve the problem step by step, we will use the information provided in the question.

### Step 1: Define Variables
Let:
- Length of the rectangle = \( L \)
- Breadth of the rectangle = \( B \)
- Radius of the circle = \( r \)

### Step 2: Use Ratios to Express Length and Breadth
Given the ratio of length to breadth is \( 7:3 \):
\[ L = 7k \quad \text{and} \quad B = 3k \]
for some constant \( k \).

### Step 3: Calculate the Perimeter of the Rectangle
The perimeter \( P \) of the rectangle is given by:
\[ P = 2(L + B) = 2(7k + 3k) = 2(10k) = 20k \]

### Step 4: Express the Circumference of the Circle
The circumference \( C \) of the circle is given to be 8 cm more than the perimeter of the rectangle:
\[ C = P + 8 = 20k + 8 \]

### Step 5: Use the Ratio of Radius to Length
The ratio of the radius of the circle to the length of the rectangle is \( 1:2 \):
\[ r = \frac{L}{2} = \frac{7k}{2} \]

### Step 6: Express the Circumference in Terms of Radius
The circumference of the circle can also be expressed as:
\[ C = 2\pi r = 2\pi \left(\frac{7k}{2}\right) = 7\pi k \]

### Step 7: Set Up the Equation
Now we can set the two expressions for circumference equal to each other:
\[ 7\pi k = 20k + 8 \]

### Step 8: Solve for \( k \)
Rearranging the equation gives:
\[ 7\pi k - 20k = 8 \]
\[ k(7\pi - 20) = 8 \]
\[ k = \frac{8}{7\pi - 20} \]

### Step 9: Find the Length of the Rectangle
Now, substitute \( k \) back into the expression for length:
\[ L = 7k = 7 \left(\frac{8}{7\pi - 20}\right) = \frac{56}{7\pi - 20} \]

### Step 10: Calculate the Length
To find the numerical value of \( L \), we can substitute \( \pi \approx 3.14 \):
\[ L = \frac{56}{7(3.14) - 20} = \frac{56}{21.98 - 20} = \frac{56}{1.98} \approx 28.28 \text{ cm} \]

### Final Answer
The length of the rectangle is approximately \( 28 \) cm.

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