If the sides of a triangular field measure 51 m, 37 m and 20 m, then find the cost of levelling it at Rs 7 per `m^(2)`.

Options

A. Rs 2562
B. Rs 2142
C. Rs 2412
D. Rs 2241

Answer

To solve the problem of finding the cost of leveling a triangular field with sides measuring 51 m, 37 m, and 20 m at a rate of Rs 7 per m², we will use Heron's formula to first calculate the area of the triangle.

### Step-by-Step Solution:

1. **Calculate the Semi-Perimeter (s)**:
The semi-perimeter \( s \) of a triangle is given by the formula:
\[ s = \frac{a + b + c}{2} \]
where \( a, b, c \) are the lengths of the sides of the triangle.
Here, \( a = 51 \, m \), \( b = 37 \, m \), and \( c = 20 \, m \).
\[ s = \frac{51 + 37 + 20}{2} = \frac{108}{2} = 54 \, m \]

**Hint**: Remember that the semi-perimeter is half the sum of the lengths of all sides.

2. **Apply Heron's Formula to Find the Area (A)**:
Heron's formula for the area of a triangle is:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Now, substitute the values:
\[ A = \sqrt{54 \times (54 - 51) \times (54 - 37) \times (54 - 20)} \]
Calculate each term:
\[ A = \sqrt{54 \times 3 \times 17 \times 34} \]

**Hint**: Break down the calculations step by step to avoid mistakes.

3. **Simplify the Expression**:
To simplify \( \sqrt{54 \times 3 \times 17 \times 34} \):
- Factor \( 54 \) as \( 6 \times 9 \) or \( 2 \times 3^3 \).
- Factor \( 34 \) as \( 2 \times 17 \).
- The expression becomes:
\[ A = \sqrt{(2 \times 3^3) \times 3 \times 17 \times (2 \times 17)} = \sqrt{2^2 \times 3^4 \times 17^2} \]
- Take the square root:
\[ A = 2 \times 3^2 \times 17 = 2 \times 9 \times 17 = 306 \, m^2 \]

**Hint**: Use properties of square roots to simplify products.

4. **Calculate the Cost of Leveling**:
The cost of leveling is given by:
\[ \text{Cost} = \text{Area} \times \text{Cost per m}^2 \]
Here, the cost per m² is Rs 7.
\[ \text{Cost} = 306 \times 7 = 2142 \, \text{Rs} \]

**Hint**: Always multiply the area by the cost per unit area to find the total cost.

### Final Answer:
The cost of leveling the triangular field is Rs 2142.