The circumference of a circular field is 440 m. A circular path of 10 metre width runs around the outside of the field. Find the cost of gravelling the path at the rate of 70 paise per square metre?

Options

A. A)Rs. 2200
B. B)Rs. 3300
C. C)Rs. 264
D. D)Can't to be determined

Answer

To solve the problem step by step, we will follow the process outlined in the video transcript while providing clear explanations for each step.

### Step 1: Find the radius of the circular field
We know the circumference (C) of the circular field is given as 440 meters. The formula for the circumference of a circle is:

\[ C = 2 \pi r \]

Where \( r \) is the radius. We can rearrange this formula to find \( r \):

\[ r = \frac{C}{2 \pi} \]

Substituting the given circumference:

\[ r = \frac{440}{2 \times \frac{22}{7}} \]

Calculating this:

\[ r = \frac{440 \times 7}{44} = \frac{3080}{44} = 70 \text{ meters} \]

### Step 2: Determine the outer radius of the path
The circular path runs around the outside of the field and has a width of 10 meters. Therefore, the outer radius \( R \) is:

\[ R = r + \text{width of the path} = 70 + 10 = 80 \text{ meters} \]

### Step 3: Calculate the area of the path
The area of the path can be calculated by finding the area of the larger circle (with radius \( R \)) and subtracting the area of the smaller circle (with radius \( r \)).

The area \( A \) of a circle is given by:

\[ A = \pi r^2 \]

Calculating the area of the larger circle:

\[ A_{\text{outer}} = \pi R^2 = \pi (80^2) = \pi (6400) \]

Calculating the area of the smaller circle:

\[ A_{\text{inner}} = \pi r^2 = \pi (70^2) = \pi (4900) \]

Now, the area of the path \( A_{\text{path}} \) is:

\[ A_{\text{path}} = A_{\text{outer}} - A_{\text{inner}} = \pi (6400 - 4900) = \pi (1500) \]

### Step 4: Calculate the area numerically
Using \( \pi \approx \frac{22}{7} \):

\[ A_{\text{path}} = \frac{22}{7} \times 1500 \]

Calculating this:

\[ A_{\text{path}} = \frac{33000}{7} \approx 4714.29 \text{ square meters} \]

### Step 5: Calculate the cost of gravelling the path
The cost of gravelling is given as 70 paise per square meter. We need to convert this to rupees:

\[ \text{Cost per square meter} = \frac{70}{100} = 0.70 \text{ rupees} \]

Now, the total cost \( C \) is:

\[ C = A_{\text{path}} \times \text{Cost per square meter} \]

Substituting the area:

\[ C = \frac{33000}{7} \times 0.70 \]

Calculating this:

\[ C = \frac{33000 \times 0.70}{7} = \frac{23100}{7} = 3300 \text{ rupees} \]

### Final Answer
The cost of gravelling the path is **3300 rupees**.

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