The cost of levelling a triangular piece of land with sides 72 m, 30 m, and 78 m is calculated at the rate of 20 paise per square metre. What is the total cost in rupees?

Introduction / Context:
This question combines triangle area calculation with a unit rate cost problem. The land is triangular, and the sides are given, so we need to use a suitable formula to find the area. Once the area is known, multiplying by the cost per square metre gives the total cost. This type of problem appears in aptitude exams to test both mensuration and basic arithmetic with money units.

Given Data / Assumptions:

• Sides of the triangle are 72 m, 30 m, and 78 m.
• The cost of levelling is 20 paise per square metre, which is ₹ 0.20 per sq. m.
• The land is a valid triangle, so the side lengths satisfy the triangle inequality.
• We must find the area of the triangle and then multiply by the rate.
• Final answer is required in rupees.

Concept / Approach:
For a triangle with all three sides known, Heron’s formula is the most direct method to find the area. According to Heron’s formula, if a, b, and c are the side lengths and s is the semi perimeter, then area = √(s(s − a)(s − b)(s − c)). After computing the area in square metres, we multiply by ₹ 0.20 per square metre to obtain the total levelling cost.

Step-by-Step Solution:
Let a = 72 m, b = 30 m, and c = 78 m.Semi perimeter s = (a + b + c) / 2 = (72 + 30 + 78) / 2 = 180 / 2 = 90 m.Heron’s formula: area = √(s(s − a)(s − b)(s − c)) = √(90 * (90 − 72) * (90 − 30) * (90 − 78)).Compute inside: 90 − 72 = 18, 90 − 30 = 60, 90 − 78 = 12, so area = √(90 * 18 * 60 * 12).90 * 18 * 60 * 12 = 1080^2, so area = 1080 sq. m. Cost = 1080 * ₹ 0.20 = ₹ 216.

Verification / Alternative check:
We can check the multiplication: 1080 * 0.20 = 1080 * (20 / 100) = (1080 * 20) / 100 = 21600 / 100 = 216. The square root step is validated because 1080^2 gives 90 * 18 * 60 * 12. Thus, there is no arithmetic slip, and the resulting cost of ₹ 216 is consistent and reasonable given the area and per metre rate.

Why Other Options Are Wrong:
Option ₹ 220 is slightly higher than the correct value and might come from rounding or misplacing a decimal point. Option ₹ 200 underestimates the total by ignoring part of the area or miscalculating the square root. Option ₹ 210 is close but still off, indicating a partial error in the area or cost multiplication. Option ₹ 240 overestimates the cost and would correspond to a larger area or higher rate, neither of which is correct.

Common Pitfalls:
Errors often arise from incorrect computation of the semi perimeter, forgetting to divide by two, or miscomputing the products inside the square root. Some students also misconvert 20 paise into rupees or accidentally use ₹ 20 per sq. m instead of ₹ 0.20 per sq. m. Careful step by step handling of both the area and the cost calculations is essential.

Final Answer:
The cost of levelling the triangular piece of land is ₹ 216.