The side of a square plot is 100 m. Two paths each of 4 m wide along the sides of the square are constructed in the middle of the plot. Find the area of the path and the cost of paving the path, if the cost of paving 10 sq. m area is Rs. 42.

Options

A. `784" m"^(2)`, Rs. 3292.80
B. `624" m"^(2)`, Rs. 3248.90
C. `624" m"^(2)`, Rs. 960.53
D. `784" m"^(2)`, Rs. 3450

Answer

To solve the problem, we need to find the area of the paths constructed in the square plot and then calculate the cost of paving those paths.

### Step 1: Calculate the area of the square plot.
The area of a square is given by the formula:
\[ \text{Area} = \text{side} \times \text{side} \]

Given that the side of the square plot is 100 m:
\[ \text{Area of the square plot} = 100 \, \text{m} \times 100 \, \text{m} = 10000 \, \text{m}^2 \]

### Step 2: Determine the dimensions of the inner square after the paths are constructed.
Since there are two paths, each 4 m wide, we need to subtract the width of the paths from each side of the square plot. The total width taken by the paths on each side is:
\[ \text{Total width of paths} = 4 \, \text{m} + 4 \, \text{m} = 8 \, \text{m} \]

Now, the side length of the inner square (the area that is not covered by the paths) will be:
\[ \text{Side of inner square} = 100 \, \text{m} - 8 \, \text{m} = 92 \, \text{m} \]

### Step 3: Calculate the area of the inner square.
Using the side length of the inner square:
\[ \text{Area of the inner square} = 92 \, \text{m} \times 92 \, \text{m} = 8464 \, \text{m}^2 \]

### Step 4: Calculate the area of the paths.
The area of the paths can be found by subtracting the area of the inner square from the area of the outer square:
\[ \text{Area of the paths} = \text{Area of the square plot} - \text{Area of the inner square} \]
\[ \text{Area of the paths} = 10000 \, \text{m}^2 - 8464 \, \text{m}^2 = 1536 \, \text{m}^2 \]

### Step 5: Calculate the cost of paving the paths.
The cost of paving is given as Rs. 42 for every 10 sq. m. First, we need to find out how many 10 sq. m are in the area of the paths:
\[ \text{Number of 10 sq. m in 1536 sq. m} = \frac{1536}{10} = 153.6 \]

Now, we multiply the number of 10 sq. m by the cost:
\[ \text{Cost of paving} = 153.6 \times 42 = 6451.2 \, \text{Rs} \]

### Final Answers:
- Area of the paths: 1536 m²
- Cost of paving the paths: Rs. 6451.2