The circumference of a circular garder is 572 m. Outside the garden a road of width 3.5 m runs around it. Calculate the cost of repairing the road at the rate of Rs 3.75 per square metre

Answer

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

### Step 1: Calculate the radius of the circular garden.
The formula for the circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]
where \( r \) is the radius. Given that the circumference of the garden is 572 m, we can rearrange the formula to find the radius:
\[ r = \frac{C}{2\pi} = \frac{572}{2\pi} \]

### Step 2: Substitute the value of \( \pi \).
Using \( \pi \approx 3.14 \):
\[ r = \frac{572}{2 \times 3.14} = \frac{572}{6.28} \approx 91.1 \text{ m} \]

### Step 3: Calculate the outer radius of the road.
The road runs around the garden with a width of 3.5 m. Therefore, the outer radius \( R \) is:
\[ R = r + 3.5 = 91.1 + 3.5 = 94.6 \text{ m} \]

### Step 4: Calculate the area of the circular garden.
The area \( A \) of a circle is given by:
\[ A = \pi r^2 \]
Substituting the radius of the garden:
\[ A_{garden} = \pi (91.1)^2 \]

### Step 5: Calculate the area of the larger circle (garden + road).
Using the outer radius:
\[ A_{outer} = \pi (94.6)^2 \]

### Step 6: Calculate the area of the road.
The area of the road is the difference between the area of the outer circle and the area of the garden:
\[ A_{road} = A_{outer} - A_{garden} \]

### Step 7: Calculate the cost of repairing the road.
The cost of repairing the road is given by the area of the road multiplied by the cost per square meter:
\[ \text{Cost} = A_{road} \times 3.75 \]

### Step 8: Substitute the values and calculate.
Now, let's perform the calculations step by step.

1. Calculate \( A_{garden} \):
\[ A_{garden} = \pi (91.1)^2 \approx 3.14 \times 8294.21 \approx 26000.14 \text{ m}^2 \]

2. Calculate \( A_{outer} \):
\[ A_{outer} = \pi (94.6)^2 \approx 3.14 \times 8962.76 \approx 28100.14 \text{ m}^2 \]

3. Calculate \( A_{road} \):
\[ A_{road} = 28100.14 - 26000.14 \approx 2100 \text{ m}^2 \]

4. Calculate the cost:
\[ \text{Cost} = 2100 \times 3.75 \approx 7875 \text{ Rs} \]

### Final Answer:
The cost of repairing the road is approximately Rs 7875.

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