The sides of a triangular farm are 20 m, 21 m and 29 m respectively. How much will be total expenses for cutting the crop at the rate of ₹ 15 per square meter?

Options

A. ₹ 3,500
B. ₹ 1,890
C. ₹ 3,150
D. ₹ 2,500

Answer

To find the total expenses for cutting the crop on a triangular farm with sides of 20 m, 21 m, and 29 m at the rate of ₹ 15 per square meter, 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 \( a, b, c \) are the sides of the triangle.

Here, \( a = 20 \, m \), \( b = 21 \, m \), and \( c = 29 \, m \).

\[ s = \frac{20 + 21 + 29}{2} = \frac{70}{2} = 35 \, m \]

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

Substituting the values:
- \( s = 35 \)
- \( a = 20 \)
- \( b = 21 \)
- \( c = 29 \)

We calculate:
\[ A = \sqrt{35(35 - 20)(35 - 21)(35 - 29)} \]
\[ A = \sqrt{35(15)(14)(6)} \]

### Step 3: Simplify the expression inside the square root.
Calculating the values:
\[ A = \sqrt{35 \times 15 \times 14 \times 6} \]

Calculating step-by-step:
1. \( 35 \times 15 = 525 \)
2. \( 14 \times 6 = 84 \)
3. \( 525 \times 84 = 44100 \)

Now, we find the square root:
\[ A = \sqrt{44100} = 210 \, m^2 \]

### Step 4: Calculate the total expenses for cutting the crop.
The cost of cutting the crop is given by:
\[ \text{Total Expenses} = \text{Area} \times \text{Rate per square meter} \]

Given the rate is ₹ 15 per square meter:
\[ \text{Total Expenses} = 210 \times 15 = 3150 \]

### Final Answer:
The total expenses for cutting the crop is ₹ 3150.

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