Around a circular garden a circular road is to be repair which costs `₹22176` at the rate of `₹1` per sq m. If the inner radius is 112 m, find the width of the circular road.

Options

A. 18 m
B. 28 m
C. 14 m
D. none of these

Answer

To find the width of the circular road around a circular garden, we can follow these steps:

### Step 1: Understand the Problem
We are given:
- The cost of repairing the circular road is ₹22176.
- The rate of repair is ₹1 per square meter.
- The inner radius of the circular garden is 112 meters.

### Step 2: Calculate the Area of the Circular Road
Since the cost is given at the rate of ₹1 per square meter, the area of the circular road can be calculated using the formula:
\[ \text{Area} = \frac{\text{Expenditure}}{\text{Rate}} = \frac{22176}{1} = 22176 \text{ sq m} \]

### Step 3: Set Up the Area Equation
Let the outer radius of the circular road be \( R \) meters. The area of the circular road can be expressed as the difference between the area of the outer circle and the area of the inner circle:
\[ \text{Area of the road} = \pi R^2 - \pi r^2 \]
Where \( r \) is the inner radius (112 m). Thus, we can write:
\[ \pi R^2 - \pi (112)^2 = 22176 \]

### Step 4: Factor Out π
Factoring out π from the left side gives us:
\[ \pi (R^2 - 112^2) = 22176 \]

### Step 5: Substitute π
Using \( \pi \approx \frac{22}{7} \):
\[ \frac{22}{7} (R^2 - 112^2) = 22176 \]

### Step 6: Clear the Fraction
Multiply both sides by 7 to eliminate the fraction:
\[ 22 (R^2 - 112^2) = 22176 \times 7 \]
Calculating the right side:
\[ 22176 \times 7 = 155232 \]
Thus, we have:
\[ 22 (R^2 - 112^2) = 155232 \]

### Step 7: Divide by 22
Now, divide both sides by 22:
\[ R^2 - 112^2 = \frac{155232}{22} \]
Calculating the division:
\[ \frac{155232}{22} = 7064 \]
So we have:
\[ R^2 - 112^2 = 7064 \]

### Step 8: Calculate 112²
Calculating \( 112^2 \):
\[ 112^2 = 12544 \]
Now, substituting this back into the equation:
\[ R^2 - 12544 = 7064 \]

### Step 9: Solve for R²
Adding \( 12544 \) to both sides gives:
\[ R^2 = 7064 + 12544 = 19608 \]

### Step 10: Take the Square Root
Now, take the square root of both sides to find \( R \):
\[ R = \sqrt{19608} = 140 \text{ m} \]

### Step 11: Calculate the Width of the Road
The width of the circular road is given by:
\[ \text{Width} = R - r = 140 - 112 = 28 \text{ m} \]

### Final Answer
The width of the circular road is **28 meters**.

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