A man walking at a speed of 4 km/h crosses a square field along its diagonal in 3 minutes. Find the area of the field in square metres.

Introduction / Context:
This problem connects speed, time, distance and area of a square. The man walks along the diagonal of a square at a known speed and takes a certain time to cross the field. From the speed and time, we get the diagonal length, from which we can determine the side length of the square and thus its area.

Given Data / Assumptions:

• Speed of man = 4 km/h
• Time taken to cross the field diagonally = 3 minutes
• Shape of field = square
• He walks along the diagonal of the square
• Relationship between diagonal and side of square: diagonal = side * sqrt(2)

Concept / Approach:
Convert speed into metres per minute so that it is compatible with the time in minutes. Multiply speed and time to get the diagonal length. Then use the relationship diagonal = side * sqrt(2) to find the side length. Finally, compute area = side^2. This chain uses basic kinematics together with geometry of squares.

Step-by-Step Solution:
Convert speed 4 km/h to metres per minute: 4 km/h = 4 * 1000 m / 60 min = 4000 / 60 ≈ 66.6667 m/min Time = 3 minutes Diagonal length d = speed * time ≈ 66.6667 * 3 ≈ 200 m For a square, diagonal d = side * sqrt(2) Let side = s, so s = d / sqrt(2) ≈ 200 / 1.414 ≈ 141.4 m Area of square = s^2 ≈ (141.4)^2 ≈ 20000 sq.m

Verification / Alternative check:
You can compute area directly using area = (diagonal^2) / 2. Since d ≈ 200 m, area = 200^2 / 2 = 40000 / 2 = 20000 sq.m exactly. This is a cleaner and exact approach and matches our approximate calculation above, confirming the result.

Why Other Options Are Wrong:
10000 and 30000 sq.m: These would correspond to incorrect side lengths or misapplied formulas. 40000 sq.m: This is diagonal^2 without dividing by 2. 25000 sq.m: Reflects a random miscalculation of the diagonal or division factor.

Common Pitfalls:
Forgetting to convert speed units properly before multiplying by time. Using perimeter formulas instead of diagonal relations for the square. Failing to square the diagonal correctly or forgetting to divide by 2.

Final Answer:
The area of the field is 20000 square metres