The area of a ring is `528cm^(2)` and the radius of the outer circle is 17 cm . Find :
(i) the radius of the smaller circle .
(ii) the width of the ring.

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript.

### Step 1: Identify the given values
- Area of the ring = 528 cm²
- Radius of the outer circle (R) = 17 cm

### Step 2: Calculate the area of the outer circle
The area of the outer circle can be calculated using the formula:
\[ \text{Area} = \pi R^2 \]
Substituting the values:
\[ \text{Area of outer circle} = \pi \times (17)^2 = \pi \times 289 \]
Using \(\pi \approx \frac{22}{7}\):
\[ \text{Area of outer circle} = \frac{22}{7} \times 289 \]
Calculating this:
\[ = \frac{22 \times 289}{7} = \frac{6358}{7} \approx 908.29 \, \text{cm}^2 \]

### Step 3: Calculate the area of the inner circle
The area of the inner circle can be found by subtracting the area of the ring from the area of the outer circle:
\[ \text{Area of inner circle} = \text{Area of outer circle} - \text{Area of ring} \]
Substituting the values:
\[ \text{Area of inner circle} = 908.29 - 528 = 380.29 \, \text{cm}^2 \]

### Step 4: Set up the equation for the area of the inner circle
The area of the inner circle can also be expressed as:
\[ \text{Area of inner circle} = \pi r^2 \]
Where \(r\) is the radius of the inner circle. Setting the two expressions for the area equal gives:
\[ \pi r^2 = 380.29 \]
Substituting \(\pi \approx \frac{22}{7}\):
\[ \frac{22}{7} r^2 = 380.29 \]

### Step 5: Solve for \(r^2\)
To isolate \(r^2\), multiply both sides by \(\frac{7}{22}\):
\[ r^2 = 380.29 \times \frac{7}{22} \]
Calculating this:
\[ r^2 = \frac{2662.03}{22} \approx 121 \]

### Step 6: Calculate \(r\)
Now, take the square root of both sides to find \(r\):
\[ r = \sqrt{121} = 11 \, \text{cm} \]

### Step 7: Calculate the width of the ring
The width of the ring is the difference between the radius of the outer circle and the radius of the inner circle:
\[ \text{Width} = R - r = 17 - 11 = 6 \, \text{cm} \]

### Final Answers:
(i) The radius of the smaller circle is **11 cm**.
(ii) The width of the ring is **6 cm**.

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