The sides of a triangular field are 140 m. 225m and 265 m . What is the cost of levelling it at Rs. 5.50 per `m^(2)` ?

Options

A. Rs. 88, 200
B. Rs. 85, 050
C. Rs. 80, 437
D. Rs. 86, 625

Answer

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

### Step 1: Calculate the semi-perimeter of the triangle
The semi-perimeter (s) is calculated using the formula:
\[ s = \frac{a + b + c}{2} \]
where \( a = 140 \, m \), \( b = 225 \, m \), and \( c = 265 \, m \).

Calculating:
\[ s = \frac{140 + 225 + 265}{2} = \frac{630}{2} = 315 \, m \]

### Step 2: Use Heron's formula to calculate the area of the triangle
Heron's formula for the area (A) of a triangle is given by:
\[ A = \sqrt{s \cdot (s - a) \cdot (s - b) \cdot (s - c)} \]

Substituting the values:
- \( s - a = 315 - 140 = 175 \)
- \( s - b = 315 - 225 = 90 \)
- \( s - c = 315 - 265 = 50 \)

Now substituting these into Heron's formula:
\[ A = \sqrt{315 \cdot 175 \cdot 90 \cdot 50} \]

### Step 3: Calculate the product inside the square root
Calculating the product:
\[ 315 \cdot 175 = 55125 \]
\[ 90 \cdot 50 = 4500 \]
Now, multiply these results:
\[ 55125 \cdot 4500 = 248062500 \]

### Step 4: Calculate the square root to find the area
Now, we take the square root:
\[ A = \sqrt{248062500} = 15750 \, m^2 \]

### Step 5: Calculate the cost of leveling the triangular field
The cost of leveling is given as Rs. 5.50 per square meter. Therefore, the total cost (C) can be calculated as:
\[ C = A \cdot \text{cost per m}^2 \]
\[ C = 15750 \cdot 5.50 \]

Calculating:
\[ C = 86525 \]

### Final Answer
The cost of leveling the triangular field is Rs. 86525.

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