The length and breadth of a park are in the ratio 2:1 and its perimeter is 240 m. A path 2 m wide runs inside it, along its boundary. Find the cost of paving the path at Rs.80 per `m^(2)`.

Answer

To solve the problem step by step, we will follow the given information and apply the necessary formulas.

### Step 1: Determine the dimensions of the park
Given that the length (L) and breadth (B) of the park are in the ratio 2:1, we can express them as:
- Let breadth = x
- Then length = 2x

### Step 2: Use the perimeter to find x
The perimeter (P) of a rectangle is given by the formula:
\[ P = 2(L + B) \]
Substituting the values we have:
\[ 240 = 2(2x + x) \]
\[ 240 = 2(3x) \]
\[ 240 = 6x \]
Now, we can solve for x:
\[ x = \frac{240}{6} = 40 \, \text{m} \]

### Step 3: Calculate the length and breadth
Now that we have x, we can find the dimensions of the park:
- Breadth (B) = x = 40 m
- Length (L) = 2x = 2(40) = 80 m

### Step 4: Calculate the area of the park
The area (A) of the park can be calculated using the formula:
\[ A = L \times B \]
\[ A = 80 \times 40 = 3200 \, \text{m}^2 \]

### Step 5: Calculate the dimensions of the inner rectangle (path)
Since there is a path of 2 m wide running along the boundary, we need to subtract 2 m from each side of the length and breadth:
- New Length = 80 - 2 - 2 = 76 m
- New Breadth = 40 - 2 - 2 = 36 m

### Step 6: Calculate the area of the inner rectangle
Now we can calculate the area of the inner rectangle:
\[ A_{inner} = 76 \times 36 \]
\[ A_{inner} = 2736 \, \text{m}^2 \]

### Step 7: Calculate the area of the path
The area of the path is the difference between the area of the park and the area of the inner rectangle:
\[ A_{path} = A - A_{inner} \]
\[ A_{path} = 3200 - 2736 = 464 \, \text{m}^2 \]

### Step 8: Calculate the cost of paving the path
The cost of paving is given as Rs. 80 per m². Therefore, the total cost can be calculated as:
\[ \text{Cost} = A_{path} \times \text{Cost per m}^2 \]
\[ \text{Cost} = 464 \times 80 = 37120 \, \text{Rs} \]

### Final Answer
The cost of paving the path is Rs. 37120.

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