The sides of a triangular park are 60 m, 112 m and 164 m.The cost of levelling the park at the rate of ₹8.50/`m^(2)` is:

Options

A. ₹18,316
B. ₹17,136
C. ₹18,164
D. ₹17,085

Answer

To solve the problem of finding the cost of levelling a triangular park with sides measuring 60 m, 112 m, and 164 m at a rate of ₹8.50 per square meter, we will follow these steps:

### Step 1: Identify the sides of the triangle
Let the sides of the triangle be:
- \( a = 60 \, \text{m} \)
- \( b = 112 \, \text{m} \)
- \( c = 164 \, \text{m} \)

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) of the triangle is calculated using the formula:
\[ s = \frac{a + b + c}{2} \]
Substituting the values:
\[ s = \frac{60 + 112 + 164}{2} = \frac{336}{2} = 168 \, \text{m} \]

### Step 3: Apply Heron's formula to find the area (A)
Heron's formula for the area of a triangle is given by:
\[ A = \sqrt{s \cdot (s - a) \cdot (s - b) \cdot (s - c)} \]
Calculating \( s - a \), \( s - b \), and \( s - c \):
- \( s - a = 168 - 60 = 108 \)
- \( s - b = 168 - 112 = 56 \)
- \( s - c = 168 - 164 = 4 \)

Now substituting these values into Heron's formula:
\[ A = \sqrt{168 \cdot 108 \cdot 56 \cdot 4} \]

### Step 4: Calculate the product inside the square root
Calculating the product:
\[ 168 \cdot 108 = 18144 \]
\[ 18144 \cdot 56 = 1016064 \]
\[ 1016064 \cdot 4 = 4064256 \]

### Step 5: Find the square root to get the area
Now, take the square root:
\[ A = \sqrt{4064256} = 2016 \, \text{m}^2 \]

### Step 6: Calculate the cost of levelling the park
The cost of levelling the park is calculated by multiplying the area by the cost per square meter:
\[ \text{Cost} = A \cdot \text{Rate} = 2016 \cdot 8.50 \]
Calculating this gives:
\[ \text{Cost} = 17136 \, \text{Rupees} \]

### Final Answer
The cost of levelling the park is ₹17136.

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