The sides of a triangular board are 13 metres, 14 metres and 15 metres. The cost of painting it at the rate of ₹8.75 per `m^2` is -----

Options

A. ₹688.80
B. ₹735
C. ₹730.80
D. ₹722.50

Answer

To find the cost of painting the triangular board with sides 13 meters, 14 meters, and 15 meters at the rate of ₹8.75 per square meter, we will first calculate the area of the triangle using Heron's formula and then determine the total cost of painting.

### Step-by-Step Solution:

1. **Identify the sides of the triangle:**
- Let \( a = 13 \) meters, \( b = 14 \) meters, and \( c = 15 \) meters.

2. **Calculate the semi-perimeter (s):**
- The semi-perimeter \( s \) is given by the formula:
\[ s = \frac{a + b + c}{2} \]
- Substituting the values:
\[ s = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21 \text{ meters} \]

3. **Calculate \( s - a \), \( s - b \), and \( s - c \):**
- \( s - a = 21 - 13 = 8 \) meters
- \( s - b = 21 - 14 = 7 \) meters
- \( s - c = 21 - 15 = 6 \) meters

4. **Apply Heron's formula to find the area (A):**
- The area \( A \) is given by:
\[ A = \sqrt{s \times (s - a) \times (s - b) \times (s - c)} \]
- Substituting the values:
\[ A = \sqrt{21 \times 8 \times 7 \times 6} \]

5. **Simplify the calculation:**
- First, calculate the product:
\[ 21 \times 8 = 168 \]
\[ 168 \times 7 = 1176 \]
\[ 1176 \times 6 = 7056 \]
- Now, take the square root:
\[ A = \sqrt{7056} = 84 \text{ m}^2 \]

6. **Calculate the cost of painting:**
- The cost of painting is given by:
\[ \text{Cost} = \text{Area} \times \text{Cost per m}^2 \]
- Substituting the values:
\[ \text{Cost} = 84 \times 8.75 \]

7. **Perform the multiplication:**
- Break it down:
\[ 84 \times 8.75 = 84 \times (8 + 0.75) = 84 \times 8 + 84 \times 0.75 \]
- Calculate:
\[ 84 \times 8 = 672 \]
\[ 84 \times 0.75 = 63 \]
- Adding these together:
\[ 672 + 63 = 735 \]

8. **Final answer:**
- The total cost of painting the triangular board is ₹735.