The ratio between the length and the breadth of a rectangular park is 3 : 2.…

When something moves at a constant speed, the distance it covers equals speed times time. For one complete lap of a closed rectangular boundary, this distance is exactly the perimeter of the rectangle, 2 times (length + breadth). If the length and breadth are in a given ratio, writing them as a common multiple of that ratio lets the perimeter equation solve for the actual side lengths, whose product then gives the area.

1. Convert the cyclist's speed to metres per minute: 12 km/hr = (12 x 1000) / 60 = 200 m/min.

2. Distance covered in one full round (the boundary/perimeter) = speed x time = 200 x 8 = 1600 m.

3. Perimeter of a rectangle = 2(length + breadth), so length + breadth = 1600 / 2 = 800 m.

4. Since length : breadth = 3 : 2, let length = 3k and breadth = 2k. Then 3k + 2k = 800, so 5k = 800 and k = 160.

5. Length = 3 x 160 = 480 m and breadth = 2 x 160 = 320 m.

6. Area = length x breadth = 480 x 320 = 153600 sq. m.

Cross-check: 2(480 + 320) = 2 x 800 = 1600 m, which matches the boundary distance found from speed and time, and 480 : 320 reduces to 3 : 2 as required, confirming the dimensions and the area of 153600 sq. m.