Assertion (A) : If the outer and inner diameter of a circular path is 10 m and 6 m then area of the path is ` 16 pi m^(2)`
Reason (R) : If R and r be the radius of outer and inner circular path . Area of circular path = `pi (R^(2) - r^(2))`

Options

A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true and R is not correct explanation of A
C. A is true but R is false
D. A is false but R is true

Answer

To solve the problem step by step, we will analyze the assertion and the reason provided.

### Step 1: Identify the outer and inner diameters
The outer diameter of the circular path is given as 10 m, and the inner diameter is given as 6 m.

### Step 2: Calculate the outer and inner radii
- The outer radius \( R \) can be calculated as:
\[ R = \frac{\text{Outer Diameter}}{2} = \frac{10 \, \text{m}}{2} = 5 \, \text{m} \]
- The inner radius \( r \) can be calculated as:
\[ r = \frac{\text{Inner Diameter}}{2} = \frac{6 \, \text{m}}{2} = 3 \, \text{m} \]

### Step 3: Use the formula for the area of the circular path
The area of the circular path can be calculated using the formula:
\[ \text{Area of circular path} = \pi (R^2 - r^2) \]

### Step 4: Substitute the values of \( R \) and \( r \)
Substituting the values of \( R \) and \( r \) into the formula:
\[ \text{Area} = \pi (5^2 - 3^2) \]
Calculating the squares:
\[ = \pi (25 - 9) \]
\[ = \pi (16) \]
\[ = 16\pi \, \text{m}^2 \]

### Step 5: Conclusion
The area of the circular path is \( 16\pi \, \text{m}^2 \), which confirms the assertion that the area of the path is \( 16\pi \, \text{m}^2 \).

### Step 6: Verify the reason
The reason states that if \( R \) and \( r \) are the radii of the outer and inner circular paths, then the area of the circular path is given by \( \pi (R^2 - r^2) \). This is indeed correct as we have used this formula to arrive at our conclusion.

### Final Answer
Both the assertion and the reason are true, and the reason correctly explains the assertion.

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