Find the area of a triangular field whose equal sides are 17 m, 15 m, and 8 m respectively . If a labour can plough `12 m^(2)` field in 1 day and gets Rs. 600 per day. Find the total labour charge he received for ploughing the field .

Answer

To solve the problem step by step, we will follow the outlined process:

### Step 1: Identify the sides of the triangle
Let the sides of the triangle be:
- \( A = 17 \, \text{m} \)
- \( B = 15 \, \text{m} \)
- \( C = 8 \, \text{m} \)

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) is calculated using the formula:
\[ s = \frac{A + B + C}{2} \]
Substituting the values:
\[ s = \frac{17 + 15 + 8}{2} = \frac{40}{2} = 20 \, \text{m} \]

### Step 3: Apply Heron's formula to find the area (A)
Heron's formula states that the area \( A \) of the triangle can be calculated as:
\[ A = \sqrt{s \cdot (s - A) \cdot (s - B) \cdot (s - C)} \]
Substituting the values:
\[ A = \sqrt{20 \cdot (20 - 17) \cdot (20 - 15) \cdot (20 - 8)} \]
Calculating each term:
\[ = \sqrt{20 \cdot 3 \cdot 5 \cdot 12} \]
Now, we can simplify:
\[ = \sqrt{20 \cdot 3 \cdot 5 \cdot 12} = \sqrt{(4 \cdot 5) \cdot 3 \cdot (4 \cdot 3)} = \sqrt{4^2 \cdot 5 \cdot 3^2} \]
Taking the square roots:
\[ = 4 \cdot 3 \cdot 5 = 60 \, \text{m}^2 \]

### Step 4: Calculate the number of days required for ploughing
Given that a laborer can plough \( 12 \, \text{m}^2 \) in one day, we need to find out how many days it will take to plough \( 60 \, \text{m}^2 \):
\[ \text{Number of days} = \frac{\text{Total area}}{\text{Area per day}} = \frac{60}{12} = 5 \, \text{days} \]

### Step 5: Calculate the total labor charge
The labor charge per day is \( 600 \, \text{Rs} \). Therefore, for 5 days:
\[ \text{Total labor charge} = \text{Number of days} \times \text{Charge per day} = 5 \times 600 = 3000 \, \text{Rs} \]

### Final Answer
The total labor charge received for ploughing the field is \( 3000 \, \text{Rs} \).

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