The perimeter of a triangular field is 540 m and its sides are in the ratio 25 : 17 : 12. Find the the area of the field. Also, find the cost of ploughing the field at Rs. 5 per `m^(2)`.

Options

A. 900 `m^(2)` Rs. 45000
B. 45000 `m^(2)` Rs. 9000
C. 9500 `m^(2)` Rs. 45000
D. 9000 `m^(2)` Rs. 45000

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Determine the sides of the triangle
Given the perimeter of the triangular field is 540 m and the sides are in the ratio 25:17:12, we can express the sides in terms of a variable \( x \).

Let the sides be:
- Side A = \( 25x \)
- Side B = \( 17x \)
- Side C = \( 12x \)

The perimeter of the triangle is the sum of all sides:
\[ A + B + C = 25x + 17x + 12x = 54x \]
Setting this equal to the given perimeter:
\[ 54x = 540 \]
Now, solve for \( x \):
\[ x = \frac{540}{54} = 10 \]

### Step 2: Calculate the lengths of the sides
Now substitute \( x \) back into the expressions for the sides:
- Side A = \( 25x = 25 \times 10 = 250 \, m \)
- Side B = \( 17x = 17 \times 10 = 170 \, m \)
- Side C = \( 12x = 12 \times 10 = 120 \, m \)

### Step 3: Calculate the semi-perimeter
The semi-perimeter \( s \) is given by:
\[ s = \frac{A + B + C}{2} = \frac{250 + 170 + 120}{2} = \frac{540}{2} = 270 \, m \]

### Step 4: Apply Heron's formula to find the area
Heron's formula states:
\[ \text{Area} = \sqrt{s(s - A)(s - B)(s - C)} \]
Substituting the values:
\[ \text{Area} = \sqrt{270(270 - 250)(270 - 170)(270 - 120)} \]
Calculating each term:
- \( s - A = 270 - 250 = 20 \)
- \( s - B = 270 - 170 = 100 \)
- \( s - C = 270 - 120 = 150 \)

Now substituting these values into the formula:
\[ \text{Area} = \sqrt{270 \times 20 \times 100 \times 150} \]

### Step 5: Simplify the area calculation
Calculating the product:
\[ 270 \times 20 = 5400 \]
\[ 5400 \times 100 = 540000 \]
\[ 540000 \times 150 = 81000000 \]
Now taking the square root:
\[ \text{Area} = \sqrt{81000000} = 9000 \, m^2 \]

### Step 6: Calculate the cost of ploughing the field
The cost of ploughing is given as Rs. 5 per \( m^2 \). Therefore, the total cost is:
\[ \text{Cost} = \text{Area} \times \text{Cost per } m^2 = 9000 \times 5 = 45000 \, Rs \]

### Final Answers
- The area of the triangular field is \( 9000 \, m^2 \).
- The cost of ploughing the field is Rs. \( 45000 \).