Calculating the Area of the Rectangular Park

The problem provides information about a rectangular park, including the ratio of its length to breadth, and the speed and time taken to cycle along its boundary. We need to find the area of the park in square meters.

Understanding the Given Information

• Ratio of Length (L) to Breadth (B) = 3 : 2. This means $\frac{L}{B} = \frac{3}{2}$, so $L = \frac{3}{2}B$ or we can say $L = 3x$ and $B = 2x$ for some variable $x$.
• Speed of cycling along the boundary = 12 km per hour.
• Time taken for one round (along the boundary) = 8 minutes.
• One round along the boundary means covering the perimeter of the park.
• We need to find the area in square meters ($\text{m}^2$).

Converting Units

The speed is given in km/hour and time in minutes, but we need the area in square meters. It's best to convert the speed and time to meters per second or meters per minute to calculate the distance (perimeter) in meters.

• Speed Conversion:

  Speed = 12 km/hour

  $1 \text{ km} = 1000 \text{ meters}$

  $1 \text{ hour} = 60 \text{ minutes} = 3600 \text{ seconds}$

  Let's convert speed to meters per minute as time is given in minutes:

  Speed = $\frac{12 \text{ km}}{1 \text{ hour}} = \frac{12 \times 1000 \text{ meters}}{60 \text{ minutes}} = \frac{12000}{60} \text{ meters/minute} = 200 \text{ meters/minute}$.

• Time Conversion:

  Time = 8 minutes.

  This is already in minutes, which is consistent with our converted speed unit.

Calculating the Perimeter of the Park

The distance covered in one round is the perimeter of the rectangular park. We can find this distance using the formula: Distance = Speed $\times$ Time.

Perimeter = Speed $\times$ Time

Perimeter = $200 \text{ meters/minute} \times 8 \text{ minutes}$

Perimeter = $1600 \text{ meters}$.

Using the Ratio to Find Length and Breadth

The perimeter of a rectangle with length L and breadth B is given by $2(L + B)$. We know the perimeter is 1600 meters.

$2(L + B) = 1600 \text{ meters}$

$L + B = \frac{1600}{2} \text{ meters}$

$L + B = 800 \text{ meters}$.

We are given the ratio $L : B = 3 : 2$. Let $L = 3x$ and $B = 2x$. Substitute these into the equation $L + B = 800$:

$3x + 2x = 800$

$5x = 800$

$x = \frac{800}{5}$

$x = 160$.

Now we can find the actual length and breadth:

• Length (L) = $3x = 3 \times 160 = 480$ meters.
• Breadth (B) = $2x = 2 \times 160 = 320$ meters.

Calculating the Area of the Park

The area of a rectangle is given by the formula: Area = Length $\times$ Breadth.

Area = $L \times B$

Area = $480 \text{ meters} \times 320 \text{ meters}$

Area = $(480 \times 320) \text{ m}^2$

Area = $153600 \text{ m}^2$.

The area of the park is 153600 square meters.

Comparing this result with the given options:

• Option 1: 15360 sq.m.
• Option 2: 153600 sq.m.
• Option 3: 30720 sq.m.
• Option 4: 307200 sq.m.

Our calculated area matches Option 2.