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 a round in 8 minutes, then the area of the park is

Options

A. `15.360m^(2)`
B. `1,53,600m^(2)`
C. `1,53,060 m^(2)`
D. `15,306m^(2)`

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the given information
The ratio of the length (L) to the breadth (B) of a rectangular park is given as 3:2. The speed of a man cycling along the boundary of the park is 12 km/hr, and he completes one round in 8 minutes.

### Step 2: Convert time from minutes to hours
Since speed is given in km/hr, we need to convert the time taken from minutes to hours.
\[ \text{Time in hours} = \frac{8 \text{ minutes}}{60} = \frac{2}{15} \text{ hours} \]

**Hint:** Remember that to convert minutes to hours, divide by 60.

### Step 3: Calculate the distance traveled
Using the formula for distance, which is speed multiplied by time, we can find the distance traveled in one round.
\[ \text{Distance} = \text{Speed} \times \text{Time} = 12 \text{ km/hr} \times \frac{2}{15} \text{ hr} \]
\[ \text{Distance} = \frac{12 \times 2}{15} = \frac{24}{15} = \frac{8}{5} \text{ km} \]

**Hint:** To find the distance, multiply the speed by the time taken.

### Step 4: Convert distance from kilometers to meters
Since 1 km = 1000 meters, we convert the distance:
\[ \text{Distance in meters} = \frac{8}{5} \text{ km} \times 1000 = \frac{8 \times 1000}{5} = 1600 \text{ meters} \]

**Hint:** Always convert units when necessary, especially when dealing with different measurements.

### Step 5: Set up the perimeter equation
The perimeter (P) of a rectangle is given by the formula:
\[ P = 2(L + B) \]
We know the perimeter is equal to the distance traveled, which is 1600 meters. Therefore:
\[ 2(L + B) = 1600 \]
\[ L + B = \frac{1600}{2} = 800 \text{ meters} \]

**Hint:** The perimeter is the total distance around the rectangle, which can be expressed in terms of length and breadth.

### Step 6: Express length and breadth in terms of x
Given the ratio of length to breadth is 3:2, we can express:
\[ L = 3x \quad \text{and} \quad B = 2x \]
Substituting these into the equation for perimeter:
\[ 3x + 2x = 800 \]
\[ 5x = 800 \]
\[ x = \frac{800}{5} = 160 \text{ meters} \]

**Hint:** Use variables to represent the unknowns based on the given ratios.

### Step 7: Calculate the dimensions of the rectangle
Now, we can find the length and breadth:
\[ L = 3x = 3 \times 160 = 480 \text{ meters} \]
\[ B = 2x = 2 \times 160 = 320 \text{ meters} \]

**Hint:** Substitute the value of x back into the expressions for length and breadth.

### Step 8: Calculate the area of the rectangle
The area (A) of the rectangle is given by:
\[ A = L \times B = 480 \times 320 \]
Calculating the area:
\[ A = 153600 \text{ square meters} \]

**Hint:** The area is found by multiplying the length by the breadth.

### Final Answer
The area of the park is **153600 m²**.