Area of a rectangular field is 3584 m and the length and the breadth are in the ratio `7 : 2`, respectively. What is the perimeter of the rectangle?

Options

A. 246 m
B. 292 m
C. 286 m
D. 288 m

Answer

To find the perimeter of the rectangular field, we can follow these steps:

### Step 1: Define the variables
Let the length of the rectangle be \( L \) and the breadth be \( B \). According to the problem, the ratio of the length to the breadth is \( 7:2 \). We can express this as:
\[ L = 7x \quad \text{and} \quad B = 2x \]
where \( x \) is a common multiplier.

### Step 2: Use the area formula
The area \( A \) of a rectangle is given by the formula:
\[ A = L \times B \]
We know the area is \( 3584 \, m^2 \). Substituting the expressions for \( L \) and \( B \):
\[ 3584 = (7x) \times (2x) \]

### Step 3: Simplify the equation
Now, simplify the equation:
\[ 3584 = 14x^2 \]

### Step 4: Solve for \( x^2 \)
To find \( x^2 \), divide both sides by 14:
\[ x^2 = \frac{3584}{14} \]
Calculating the right side:
\[ x^2 = 256 \]

### Step 5: Solve for \( x \)
Now, take the square root of both sides to find \( x \):
\[ x = \sqrt{256} = 16 \]

### Step 6: Find the length and breadth
Now substitute \( x \) back into the expressions for \( L \) and \( B \):
\[ L = 7x = 7 \times 16 = 112 \, m \]
\[ B = 2x = 2 \times 16 = 32 \, m \]

### Step 7: Calculate the perimeter
The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2(L + B) \]
Substituting the values of \( L \) and \( B \):
\[ P = 2(112 + 32) = 2 \times 144 = 288 \, m \]

### Final Answer
The perimeter of the rectangle is \( 288 \, m \).

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