The difference in the areas of two concentric circles is 66 `cm^2` and the radius of the outer circle is 11 cm. What is the radius of the inner circle?

Options

A. 8 cm
B. 9 cm
C. 10 cm
D. 7 cm

Answer

To find the radius of the inner circle, we can follow these steps:

### Step 1: Understand the formula for the area of a circle
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.

### Step 2: Calculate the area of the outer circle
Given that the radius of the outer circle is 11 cm, we can calculate its area:
\[ A_{\text{outer}} = \pi (11)^2 = \pi \times 121 = 121\pi \, \text{cm}^2 \]

### Step 3: Set up the equation for the area of the inner circle
Let the radius of the inner circle be \( r \) cm. The area of the inner circle is:
\[ A_{\text{inner}} = \pi r^2 \]

### Step 4: Write the equation for the difference in areas
According to the problem, the difference in the areas of the two circles is 66 cm²:
\[ A_{\text{outer}} - A_{\text{inner}} = 66 \]
Substituting the areas we calculated:
\[ 121\pi - \pi r^2 = 66 \]

### Step 5: Simplify the equation
We can factor out \( \pi \) from the left side:
\[ \pi (121 - r^2) = 66 \]
Now, divide both sides by \( \pi \):
\[ 121 - r^2 = \frac{66}{\pi} \]

### Step 6: Solve for \( r^2 \)
Rearranging the equation gives:
\[ r^2 = 121 - \frac{66}{\pi} \]

### Step 7: Calculate \( r^2 \) using the approximate value of \( \pi \)
Using \( \pi \approx 3.14 \):
\[ r^2 = 121 - \frac{66}{3.14} \approx 121 - 21.05 \approx 99.95 \]

### Step 8: Find \( r \)
Now, take the square root of both sides to find \( r \):
\[ r \approx \sqrt{99.95} \approx 10 \, \text{cm} \]

### Conclusion
The radius of the inner circle is approximately 10 cm.

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