The ratio between the perimeter and the breadth of a rectangle is 3 : 1. If the area of the rectangle is 310 sq. cm, the length of the rectangle is nearly:

Options

A. `11.45` cm
B. `10.45` cm
C. `12.45` cm
D. `13.45` cm

Answer

To find the length of the rectangle given the ratio of the perimeter to the breadth and the area, we can follow these steps:

### Step 1: Understand the given information
We know:
- The ratio of the perimeter (P) to the breadth (b) is 3:1.
- The area (A) of the rectangle is 310 sq. cm.

### Step 2: Express the perimeter in terms of length (l) and breadth (b)
The formula for the perimeter of a rectangle is:
\[ P = 2(l + b) \]

### Step 3: Set up the ratio equation
Given the ratio of the perimeter to the breadth is 3:1, we can express this as:
\[ \frac{P}{b} = 3 \]
Substituting the expression for perimeter:
\[ \frac{2(l + b)}{b} = 3 \]

### Step 4: Simplify the equation
Multiplying both sides by b:
\[ 2(l + b) = 3b \]
Now, distribute the 2:
\[ 2l + 2b = 3b \]

### Step 5: Solve for length in terms of breadth
Rearranging the equation gives:
\[ 2l = 3b - 2b \]
\[ 2l = b \]
Thus, we have:
\[ l = \frac{b}{2} \]

### Step 6: Use the area to find the dimensions
The area of a rectangle is given by:
\[ A = l \times b \]
Substituting the area value:
\[ 310 = l \times b \]
Now substitute \( l \) from the previous step:
\[ 310 = \left(\frac{b}{2}\right) \times b \]
This simplifies to:
\[ 310 = \frac{b^2}{2} \]

### Step 7: Solve for breadth
Multiplying both sides by 2 gives:
\[ 620 = b^2 \]
Taking the square root of both sides:
\[ b = \sqrt{620} \]

### Step 8: Calculate breadth
Calculating the square root:
\[ b \approx 24.9 \text{ cm} \]

### Step 9: Find the length using the breadth
Now substitute back to find the length:
\[ l = \frac{b}{2} = \frac{24.9}{2} \approx 12.45 \text{ cm} \]

### Final Answer
The length of the rectangle is approximately **12.45 cm**.

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