A park is of the shape of a circle of diameter 7 m. It is surrounded by a path of width of 0.7 m. Find the expenditure of cementing the path. If its cost is Rs. 110 per sq. m.

Answer

To solve the problem, we need to find the area of the path surrounding the circular park and then calculate the cost of cementing that area. Here’s a step-by-step solution:

### Step 1: Find the radius of the park
The diameter of the park is given as 7 m. The radius (r) is half of the diameter.
\[ r = \frac{\text{Diameter}}{2} = \frac{7}{2} = 3.5 \text{ m} \]

### Step 2: Find the radius of the outer circle (park + path)
The path has a width of 0.7 m. Therefore, the radius of the outer circle (R) is the radius of the park plus the width of the path.
\[ R = r + \text{Width of the path} = 3.5 + 0.7 = 4.2 \text{ m} \]

### Step 3: Calculate the area of the park
The area of a circle is given by the formula \( A = \pi r^2 \). We will calculate the area of the park using its radius.
\[ \text{Area of the park} = \pi r^2 = \pi (3.5)^2 = \pi \times 12.25 \approx 38.48 \text{ m}^2 \]

### Step 4: Calculate the area of the outer circle
Now, we will calculate the area of the outer circle using its radius.
\[ \text{Area of the outer circle} = \pi R^2 = \pi (4.2)^2 = \pi \times 17.64 \approx 55.41 \text{ m}^2 \]

### Step 5: Calculate the area of the path
The area of the path is the difference between the area of the outer circle and the area of the park.
\[ \text{Area of the path} = \text{Area of the outer circle} - \text{Area of the park} \approx 55.41 - 38.48 \approx 16.93 \text{ m}^2 \]

### Step 6: Calculate the cost of cementing the path
The cost of cementing is given as Rs. 110 per square meter. Therefore, the total cost can be calculated as:
\[ \text{Cost} = \text{Area of the path} \times \text{Cost per sq. m} = 16.93 \times 110 \approx 1862.3 \]

### Final Answer
The expenditure of cementing the path is approximately Rs. 1862.30.

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