A triangular board has sides 13 m, 14 m, and 15 m. Find the cost of painting it at Rs 8.75 per square meter.

Introduction / Context:
To price painting a triangular board, first compute its area, then multiply by the given rate per square meter. For sides 13, 14, and 15, Heron’s formula gives a clean, integer area, making the cost straightforward to evaluate.

Given Data / Assumptions:

• Sides: a = 13 m, b = 14 m, c = 15 m
• Rate = Rs 8.75 per m^2
• Heron’s formula: Area = √(s(s − a)(s − b)(s − c)) where s = (a + b + c)/2

Concept / Approach:
Compute semi-perimeter s, plug values into Heron’s formula to get exact area (no rounding), then multiply by the rate to obtain cost.

Step-by-Step Solution:
s = (13 + 14 + 15)/2 = 42/2 = 21Area = √(21 * (21 − 13) * (21 − 14) * (21 − 15)) = √(21 * 8 * 7 * 6) = √7056 = 84 m^2Cost = 84 * 8.75 = Rs 735

Verification / Alternative check:
Because 13, 14, 15 is a near-Pythagorean set, Heron yields an integer area (84). Multiplying by 8.75 produces an exact currency amount without decimals, as seen here.

Why Other Options Are Wrong:
Rs 688.80, Rs 730.80, and Rs 722.50 correspond to different (incorrect) areas or rates; only Rs 735 matches area 84 m^2 at Rs 8.75/m^2.

Common Pitfalls:
Arithmetic slips when computing the square root in Heron’s formula, or rounding intermediate steps leading to off-by-a-few-rupees totals. Keep exact integers to the end.

Final Answer:
Rs. 735