A rectangular plot of land measures 95 m by 72 m. Inside the plot, a path of uniform width of 3.5 m is to be constructed all around. The rest of the plot is to be laid with grass. Find the total expenses involved in constructing the path at Rs.80 per m and laying the grass at Rs.40 per `m^(2)`.

Answer

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

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

Given:
- Length = 95 m
- Width = 72 m

Calculating the area:
\[ \text{Area} = 95 \, \text{m} \times 72 \, \text{m} = 6840 \, \text{m}^2 \]

### Step 2: Calculate the dimensions of the inner rectangle (area of grass).
The path has a uniform width of 3.5 m around the plot. Therefore, we need to subtract twice the width of the path (3.5 m on each side) from both the length and width of the plot.

Calculating the new dimensions:
- New Length = \( 95 \, \text{m} - 2 \times 3.5 \, \text{m} = 95 \, \text{m} - 7 \, \text{m} = 88 \, \text{m} \)
- New Width = \( 72 \, \text{m} - 2 \times 3.5 \, \text{m} = 72 \, \text{m} - 7 \, \text{m} = 65 \, \text{m} \)

Now, calculate the area of the inner rectangle:
\[ \text{Area of grass} = 88 \, \text{m} \times 65 \, \text{m} = 5720 \, \text{m}^2 \]

### Step 3: Calculate the area of the path.
The area of the path can be found by subtracting the area of the grass from the area of the entire plot.

Calculating the area of the path:
\[ \text{Area of path} = \text{Area of plot} - \text{Area of grass} \]
\[ \text{Area of path} = 6840 \, \text{m}^2 - 5720 \, \text{m}^2 = 1120 \, \text{m}^2 \]

### Step 4: Calculate the cost of constructing the path.
The cost of constructing the path is given as Rs. 80 per meter of path. The perimeter of the path can be calculated as follows:

Calculating the perimeter of the outer rectangle:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) = 2 \times (95 \, \text{m} + 72 \, \text{m}) = 2 \times 167 \, \text{m} = 334 \, \text{m} \]

Now, calculating the cost:
\[ \text{Cost of path} = \text{Perimeter} \times \text{Cost per meter} = 334 \, \text{m} \times 80 \, \text{Rs/m} = 26720 \, \text{Rs} \]

### Step 5: Calculate the cost of laying the grass.
The cost of laying grass is Rs. 40 per square meter.

Calculating the cost:
\[ \text{Cost of grass} = \text{Area of grass} \times \text{Cost per square meter} = 5720 \, \text{m}^2 \times 40 \, \text{Rs/m}^2 = 228800 \, \text{Rs} \]

### Step 6: Calculate the total expenses.
Now, we can find the total expenses by adding the cost of the path and the cost of laying the grass.

Calculating total expenses:
\[ \text{Total Expenses} = \text{Cost of path} + \text{Cost of grass} = 26720 \, \text{Rs} + 228800 \, \text{Rs} = 255520 \, \text{Rs} \]

### Final Answer:
The total expenses involved in constructing the path and laying the grass is Rs. 255520.

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