A cow is tethered by a rope so that it can graze exactly over an area of 2826 square metres. Assuming the grazing region is a perfect circle and the rope represents the radius, what is the length of the rope in metres?

Introduction:
This is a straightforward application of the formula for the area of a circle. The situation describes a cow tied to a fixed point with a rope, and the cow can graze in a circular region around that point. The area grazed is given, and we are asked to determine the radius of the circle, which is the length of the rope. Such questions often appear in quantitative aptitude to test comfort with circle geometry, basic formula manipulation, and arithmetic.

Given Data / Assumptions:

• The grazing area is circular. • Area of the circle grazed by the cow = 2826 square metres. • The radius of the circle is equal to the length of the rope. • Use the standard formula: area of circle = pi * r^2. • Use pi = 3.14 for calculations.

Concept / Approach:
The area A of a circle is related to its radius r by A = pi * r^2. When A is known, we can rearrange the formula to solve for r: r^2 = A / pi and r = sqrt(A / pi). The problem becomes a simple substitution and square root calculation. Once the radius is obtained, we interpret it directly as the rope length in metres.

Step-by-Step Solution:
Step 1: Write the formula for the area of a circle. A = pi * r^2. Step 2: Substitute the given area A = 2826 sq m and pi = 3.14. 2826 = 3.14 * r^2. Step 3: Solve for r^2. r^2 = 2826 / 3.14. Step 4: Perform the division. 2826 / 3.14 = 900. Step 5: Take the square root of r^2. r = sqrt(900) = 30. Thus, the radius of the circular grazing area, and therefore the length of the rope, is 30 metres.

Verification / Alternative check:
Check by substitution: if r = 30 metres, area = pi * r^2 = 3.14 * 30 * 30 = 3.14 * 900 = 2826 square metres. This exactly matches the given grazing area, confirming that the radius value is correct. There is no ambiguity in the shape or formula used, so this verification is strong evidence that 30 metres is the correct rope length.

Why Other Options Are Wrong:
Option 15 m: Using this as radius gives area = 3.14 * 15 * 15 = 706.5 sq m, which is far smaller than 2826 sq m. Option 24 m: Area would be 3.14 * 24 * 24 = 3.14 * 576 = 1808.64 sq m, still much less than required. Option 36 m: Area would be 3.14 * 36 * 36 = 3.14 * 1296 ≈ 4069.44 sq m, which is significantly larger than the given area. Option 20 m: Area would be 3.14 * 20 * 20 = 1256 sq m, again smaller than the required 2826 sq m.

Common Pitfalls:
Students sometimes confuse radius and diameter and may accidentally set 2r^2 instead of r^2, or they may forget to take the square root after dividing the area by pi. Some also mistakenly plug the area into pi * d^2 / 4 directly without careful substitution, which complicates the calculation. The safest route is always to start from A = pi * r^2, rearrange clearly, and proceed with systematic substitution and simplification.

Final Answer:
The length of the rope is 30 metres.