The area of a circular cricket ground is 24.64 hectares. Find the cost of making a rope boundary all around the ground at the rate of ₹5.40 per metre.

Introduction / Context: This problem links area and circumference of a circle with a real life application involving cost. You are given the area of a circular cricket ground in hectares and asked to compute the cost of placing a rope boundary around it. The steps involve unit conversion, finding the radius from the area, calculating the circumference and then multiplying by the given rate per metre.

Given Data / Assumptions:

• Area of the circular ground A = 24.64 hectares.
• 1 hectare = 10,000 square metres.
• Rate of rope per metre = ₹5.40.
• We assume π ≈ 3.14 for circular calculations.
• Area of a circle A = πr^2, circumference C = 2πr.

Concept / Approach: First convert the area from hectares to square metres. Then use the area formula A = πr^2 to find the radius r. After that, find the circumference C = 2πr, which represents the length of the rope required to surround the ground. Finally, calculate the total cost by multiplying C by the given rate per metre. Approximations using π = 3.14 are standard for such multiple choice questions.

Step-by-Step Solution: Convert area to square metres: A = 24.64 hectares = 24.64 * 10000 = 246400 sq m. Use A = πr^2 with π ≈ 3.14: 246400 ≈ 3.14 * r^2. So r^2 ≈ 246400 / 3.14. Compute r^2 ≈ 78471.3 (approximate value). Take square root: r ≈ 280 m (approximately 280.1 m). Circumference C ≈ 2 * 3.14 * 280 ≈ 1759 m. Cost = C * rate ≈ 1759 * 5.40 ≈ ₹9504 (approximate).

Verification / Alternative check: The key is to be consistent with the approximations. Using r ≈ 280.1 m rather than exactly 280 m produces nearly the same circumference and thus cost. The values cluster very close around ₹9500, and among the options, ₹9,504 is the closest and clearly intended answer. Small rounding differences are normal in such questions and are accounted for in the options provided.

Why Other Options Are Wrong: ₹9,600: Slightly larger and would correspond to a slightly bigger radius than justified by the given area. ₹9,802 and ₹9,876: These values are further away from the consistent rounding with π ≈ 3.14 and the given area. ₹9,300: Too low compared to the carefully calculated approximate cost.

Common Pitfalls: Forgetting to convert hectares to square metres before applying the area formula. Using an incorrect value of π or inconsistent rounding that leads to large errors. Mistakes in taking the square root of r^2 or in the final multiplication with the rate per metre.

Final Answer: The cost of making the rope boundary is approximately ₹9,504