The areas of two concentric circles are `962.5 cm^(2)` and `1386 cm^(2)` respectively. The width of the ring is ___________.

Options

A. 3.4 cm
B. 3.5 cm
C. 3.2 cm
D. 3.1 cm

Answer

To find the width of the ring formed by two concentric circles with given areas, we can follow these steps:

### Step 1: Understand the problem
We have two concentric circles with areas given as:
- Area of the smaller circle (A) = 962.5 cm²
- Area of the larger circle (B) = 1386 cm²

### Step 2: Use 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 3: Calculate the radius of the larger circle
For the larger circle (B):
\[ \pi r_2^2 = 1386 \]
Using \( \pi \approx \frac{22}{7} \):
\[ \frac{22}{7} r_2^2 = 1386 \]
To isolate \( r_2^2 \), multiply both sides by \( \frac{7}{22} \):
\[ r_2^2 = 1386 \times \frac{7}{22} \]
Calculating the right side:
\[ r_2^2 = 1386 \times \frac{7}{22} = 1386 \times 0.31818 \approx 441 \]
Now, take the square root to find \( r_2 \):
\[ r_2 = \sqrt{441} = 21 \text{ cm} \]

### Step 4: Calculate the radius of the smaller circle
For the smaller circle (A):
\[ \pi r_1^2 = 962.5 \]
Using \( \pi \approx \frac{22}{7} \):
\[ \frac{22}{7} r_1^2 = 962.5 \]
To isolate \( r_1^2 \), multiply both sides by \( \frac{7}{22} \):
\[ r_1^2 = 962.5 \times \frac{7}{22} \]
Calculating the right side:
\[ r_1^2 = 962.5 \times \frac{7}{22} = 962.5 \times 0.31818 \approx 305.25 \]
Now, take the square root to find \( r_1 \):
\[ r_1 = \sqrt{305.25} \approx 17.5 \text{ cm} \]

### Step 5: Calculate the width of the ring
The width of the ring is given by the difference between the radii of the larger and smaller circles:
\[ \text{Width} = r_2 - r_1 = 21 \text{ cm} - 17.5 \text{ cm} = 3.5 \text{ cm} \]

### Final Answer
The width of the ring is **3.5 cm**.

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