Ina circular garden of radius 15 m, a path of 2 m wide has to be made inside the garden at the rate of ₹ 24 per sq. m. The cost of making the path is: ( Take `pi=22/7` )

Options

A. Rs 4100
B. Rs 3864
C. Rs 4224
D. Rs 4355

Answer

To solve the problem, we need to calculate the area of the path inside the circular garden and then find the cost of making that path. Here’s a step-by-step breakdown:

### Step 1: Identify the radii
The radius of the circular garden is given as 15 m. The path is 2 m wide, so we need to find the radius of the inner circle (the area without the path).

- **Outer radius (R1)** = 15 m
- **Inner radius (R2)** = 15 m - 2 m = 13 m

### Step 2: Calculate the area of the outer circle
The area of a circle is given by the formula:
\[ \text{Area} = \pi r^2 \]

For the outer circle:
\[ \text{Area}_{\text{outer}} = \pi (R1)^2 = \pi (15)^2 = \pi (225) \]

Using \(\pi = \frac{22}{7}\):
\[ \text{Area}_{\text{outer}} = \frac{22}{7} \times 225 \]

### Step 3: Calculate the area of the inner circle
For the inner circle:
\[ \text{Area}_{\text{inner}} = \pi (R2)^2 = \pi (13)^2 = \pi (169) \]

Using \(\pi = \frac{22}{7}\):
\[ \text{Area}_{\text{inner}} = \frac{22}{7} \times 169 \]

### Step 4: 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 inner circle:
\[ \text{Area}_{\text{path}} = \text{Area}_{\text{outer}} - \text{Area}_{\text{inner}} \]

Substituting the areas calculated:
\[ \text{Area}_{\text{path}} = \left(\frac{22}{7} \times 225\right) - \left(\frac{22}{7} \times 169\right) \]
\[ = \frac{22}{7} \times (225 - 169) \]
\[ = \frac{22}{7} \times 56 \]

### Step 5: Calculate the area of the path
Now, calculate:
\[ \text{Area}_{\text{path}} = \frac{22 \times 56}{7} \]
\[ = \frac{1232}{7} \]
\[ = 176 \, \text{m}^2 \]

### Step 6: Calculate the cost of making the path
The cost of making the path is given as ₹ 24 per square meter. Therefore, the total cost is:
\[ \text{Cost} = \text{Area}_{\text{path}} \times \text{Rate} \]
\[ = 176 \times 24 \]

### Step 7: Calculate the total cost
Now, calculate:
\[ \text{Cost} = 4224 \, \text{Rupees} \]

### Final Answer
The cost of making the path is ₹ 4224.

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