The diagonal of a rectangle field is 15 m and its area is 108 `m^2` . What is the cost of fencing the field at ₹50.50 per m?

Options

A. ₹2121
B. ₹2020
C. ₹1919
D. ₹2222

Answer

To solve the problem step by step, we will use the given information about the rectangle's diagonal and area to find the cost of fencing the field.

### Step 1: Understand the given information
We know:
- The diagonal (d) of the rectangle is 15 m.
- The area (A) of the rectangle is 108 m².

### Step 2: Set up the equations
Let the length of the rectangle be \( l \) and the breadth be \( b \).
From the properties of rectangles, we have:
1. The area:
\[ l \times b = 108 \quad \text{(1)} \]
2. The diagonal:
\[ d = \sqrt{l^2 + b^2} \implies 15 = \sqrt{l^2 + b^2} \quad \text{(2)} \]

### Step 3: Square the diagonal equation
Squaring equation (2) gives:
\[ 15^2 = l^2 + b^2 \implies 225 = l^2 + b^2 \quad \text{(3)} \]

### Step 4: Use equations (1) and (3) to find \( l + b \)
We can use the identity:
\[ (l + b)^2 = l^2 + b^2 + 2lb \]
Substituting from equations (3) and (1):
\[ (l + b)^2 = 225 + 2 \times 108 \]
Calculating \( 2 \times 108 = 216 \):
\[ (l + b)^2 = 225 + 216 = 441 \]

### Step 5: Find \( l + b \)
Taking the square root:
\[ l + b = \sqrt{441} = 21 \quad \text{(4)} \]

### Step 6: Calculate the perimeter
The perimeter \( P \) of the rectangle is given by:
\[ P = 2(l + b) \]
Substituting from equation (4):
\[ P = 2 \times 21 = 42 \text{ m} \]

### Step 7: Calculate the cost of fencing
The cost of fencing at ₹50.50 per meter is:
\[ \text{Cost} = P \times \text{cost per meter} = 42 \times 50.50 \]
Calculating:
\[ 42 \times 50.50 = 2111 \]

### Final Answer
The cost of fencing the field is ₹2111.

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