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

Options

A. Rs 45000
B. Rs 50000
C. Rs 48500
D. Rs 42500

Explanation

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

### Step 1: Determine the sides of the triangle
Given the ratio of the sides of the triangle is 25:17:12, we can express the sides as:
- Side A = 25x
- Side B = 17x
- Side C = 12x

The perimeter of the triangle is given as 540 m. Therefore, we can write the equation:
\[ 25x + 17x + 12x = 540 \]

### Step 2: Simplify the equation
Combine the terms on the left side:
\[ 54x = 540 \]

### Step 3: Solve for x
To find the value of x, divide both sides by 54:
\[ x = \frac{540}{54} = 10 \]

### Step 4: Calculate the lengths of the sides
Now that we have the value of x, we can find the lengths of 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 5: Calculate the semi-perimeter
The semi-perimeter (s) is given by:
\[ s = \frac{\text{Perimeter}}{2} = \frac{540}{2} = 270 \, \text{m} \]

### Step 6: Use Heron's formula to find the area
Heron's formula states that the area (A) of a triangle can be calculated as:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Where:
- \( a = 250 \, m \)
- \( b = 170 \, m \)
- \( c = 120 \, m \)

Substituting the values into Heron's formula:
\[ A = \sqrt{270(270 - 250)(270 - 170)(270 - 120)} \]
\[ A = \sqrt{270(20)(100)(150)} \]

### Step 7: Calculate the area
Calculate the values inside the square root:
\[ A = \sqrt{270 \times 20 \times 100 \times 150} \]
\[ A = \sqrt{81000000} \]
\[ A = 9000 \, m^2 \]

### Step 8: Calculate the cost of ploughing
The cost of ploughing is given as Rs 5 per m². Therefore, the total cost (C) is:
\[ C = \text{Area} \times \text{Cost per m²} \]
\[ C = 9000 \times 5 = 45000 \, \text{Rs} \]

### Final Answer
The cost of ploughing the field is Rs 45000.

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