The diagonal of a rectangular field is 18 m and its area is `126 m^(2)`. What will be the total expenditure is fencing the field at the rate of 9 per metre ?

Options

A. Rs. 432
B. Rs. 420
C. Rs. 442
D. Rs. 430

Answer

To solve the problem step by step, we need to find the dimensions of the rectangular field first, then calculate the perimeter, and finally determine the total expenditure for fencing.

### Step 1: Use the given area to set up an equation
Let the length of the rectangle be \( l \) meters and the width be \( w \) meters. We know that:
\[ l \times w = 126 \quad \text{(1)} \]

### Step 2: Use the Pythagorean theorem for the diagonal
The diagonal \( d \) of the rectangle can be expressed using the Pythagorean theorem:
\[ d^2 = l^2 + w^2 \]
Given that the diagonal is 18 meters, we have:
\[ 18^2 = l^2 + w^2 \]
This simplifies to:
\[ 324 = l^2 + w^2 \quad \text{(2)} \]

### Step 3: Solve the system of equations
Now we have two equations:
1. \( l \times w = 126 \) (from equation (1))
2. \( l^2 + w^2 = 324 \) (from equation (2))

From equation (1), we can express \( w \) in terms of \( l \):
\[ w = \frac{126}{l} \]

Substituting this into equation (2):
\[ l^2 + \left(\frac{126}{l}\right)^2 = 324 \]

### Step 4: Simplify and solve for \( l \)
This leads to:
\[ l^2 + \frac{15876}{l^2} = 324 \]
Multiplying through by \( l^2 \) to eliminate the fraction:
\[ l^4 - 324l^2 + 15876 = 0 \]
Letting \( x = l^2 \), we have a quadratic equation:
\[ x^2 - 324x + 15876 = 0 \]

### Step 5: Use the quadratic formula to find \( x \)
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -324, c = 15876 \):
\[ x = \frac{324 \pm \sqrt{(-324)^2 - 4 \cdot 1 \cdot 15876}}{2 \cdot 1} \]
Calculating the discriminant:
\[ (-324)^2 = 104976 \]
\[ 4 \cdot 1 \cdot 15876 = 63504 \]
Thus,
\[ x = \frac{324 \pm \sqrt{104976 - 63504}}{2} \]
\[ x = \frac{324 \pm \sqrt{41352}}{2} \]
Calculating \( \sqrt{41352} \):
\[ \sqrt{41352} = 203.36 \quad \text{(approximately)} \]
So,
\[ x = \frac{324 \pm 203.36}{2} \]

### Step 6: Calculate the possible values for \( x \)
Calculating the two possible values:
1. \( x = \frac{324 + 203.36}{2} = 263.68 \)
2. \( x = \frac{324 - 203.36}{2} = 60.32 \)

### Step 7: Find \( l \) and \( w \)
Taking square roots:
1. \( l^2 = 263.68 \Rightarrow l \approx 16.24 \)
2. \( w^2 = 60.32 \Rightarrow w \approx 7.77 \)

### Step 8: Calculate the perimeter
The perimeter \( P \) of the rectangle is given by:
\[ P = 2(l + w) = 2(16.24 + 7.77) = 2(24.01) \approx 48.02 \text{ meters} \]

### Step 9: Calculate the total expenditure
The cost of fencing is given as 9 per meter:
\[ \text{Total expenditure} = P \times 9 = 48.02 \times 9 \approx 432.18 \]

### Final Answer
The total expenditure for fencing the field is approximately **432.18**.