The difference between the radii of the bigger circle and smaller circle is 14 cm and the difference between their areas is `1056 cm^(2)`. Radius of the smaller circle is

Options

A. 7 cm
B. 5 cm
C. 9 cm
D. 3 cm

Answer

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

### Step 1: Define Variables
Let the radius of the smaller circle be \( r \) cm.
Then, the radius of the bigger circle will be \( r + 14 \) cm.

### Step 2: Write the Area Difference Equation
The area of a circle is given by the formula \( A = \pi r^2 \).
The area of the smaller circle is \( \pi r^2 \) and the area of the bigger circle is \( \pi (r + 14)^2 \).
The difference between the areas of the two circles is given as 1056 cm².
So, we can write the equation:
\[ \pi (r + 14)^2 - \pi r^2 = 1056 \]

### Step 3: Factor Out \( \pi \)
Factoring out \( \pi \) from the left side gives:
\[ \pi \left((r + 14)^2 - r^2\right) = 1056 \]
Dividing both sides by \( \pi \):
\[ (r + 14)^2 - r^2 = \frac{1056}{\pi} \]

### Step 4: Simplify the Left Side
Using the difference of squares:
\[ (r + 14 - r)(r + 14 + r) = \frac{1056}{\pi} \]
This simplifies to:
\[ 14(2r + 14) = \frac{1056}{\pi} \]

### Step 5: Substitute \( \pi \)
Using \( \pi \approx \frac{22}{7} \):
\[ 14(2r + 14) = \frac{1056 \times 7}{22} \]
Calculating the right side:
\[ 14(2r + 14) = \frac{7392}{22} \]
\[ 14(2r + 14) = 336 \]

### Step 6: Solve for \( r \)
Now, divide both sides by 14:
\[ 2r + 14 = \frac{336}{14} \]
\[ 2r + 14 = 24 \]
Subtract 14 from both sides:
\[ 2r = 10 \]
Now divide by 2:
\[ r = 5 \]

### Conclusion
The radius of the smaller circle is \( \mathbf{5 \, cm} \).

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