The ratio between the length and the breadth of a rectangular park is 3 : 2. If a man cycling along the boundary of the park at the speed of 12 km/hr completes one round in 8 minutes, then the area of the park (in sq. m) is

### Concept & Distance Cycling along the boundary of the park means covering the perimeter. First, calculate the distance covered using speed and time, which gives the perimeter. $$ \text{Distance} = \text{Speed} \times \text{Time} $$ ### Step-by-Step Solution * Convert the time into hours to match the speed units: $$ \text{Time} = \frac{8}{60} \text{ hours} = \frac{2}{15} \text{ hours} $$ * Calculate the distance (perimeter) covered in km: $$ \text{Perimeter} = 12 \text{ km/hr} \times \frac{2}{15} \text{ hr} = \frac{24}{15} \text{ km} = 1.6 \text{ km} $$ * Convert the perimeter to meters (since the final area needs to be in sq. m): $$ 1.6 \text{ km} = 1600 \text{ m} $$ * Use the ratio $3:2$ to find the dimensions. Let length be $3x$ and breadth be $2x$. $$ 2(3x + 2x) = 1600 $$ $$ 2(5x) = 1600 \implies 10x = 1600 \implies x = 160 $$ * Calculate the actual sides: * Length = $3 \times 160 = 480 \text{ m}$ * Breadth = $2 \times 160 = 320 \text{ m}$ * Calculate the area: $$ \text{Area} = 480 \times 320 = 153600 \text{ sq. m} $$ ### Exam Strategy & Shortcut To quickly convert $12 \text{ km/hr}$ to $\text{m/s}$, multiply by $5/18$, yielding $10/3 \text{ m/s}$. The time is 8 minutes = $480 \text{ s}$. Perimeter = $(10/3) \times 480 = 1600 \text{ m}$. The ratio is $3:2$, so perimeter is $10$ parts. 1 part = 160. Area = $3 \text{ parts} \times 2 \text{ parts} = 6 \text{ parts}^2 = 6 \times (160)^2 = 6 \times 25600 = 153600$. ### Common Pitfall Failing to convert the speed from km/hr to m/min or time from minutes to hours, leading to a massive miscalculation in the perimeter size. ### Final Answer Therefore, the correct answer is **153600 sq. m**.