The area enclosed by the circumference of two concentric circles is `423.5 cm^(2)`. If the circumference of outer circle is 132 cm. Calculate the radius of the inner circle

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Understand the Problem
We have two concentric circles (circles with the same center) and we know the area between them and the circumference of the outer circle. We need to find the radius of the inner circle.

### Step 2: Given Information
- Area enclosed by the two circles = 423.5 cm²
- Circumference of the outer circle = 132 cm

### Step 3: Find the Outer Radius (R₀)
The formula for the circumference of a circle is given by:
\[ C = 2 \pi R \]
For the outer circle, we can write:
\[ 132 = 2 \pi R₀ \]
Substituting \(\pi\) with \(\frac{22}{7}\):
\[ 132 = 2 \times \frac{22}{7} \times R₀ \]
Now, simplify this equation to find \(R₀\):
\[ 132 = \frac{44}{7} R₀ \]
Multiplying both sides by \(7\):
\[ 924 = 44 R₀ \]
Now, divide both sides by \(44\):
\[ R₀ = \frac{924}{44} = 21 \text{ cm} \]

### Step 4: Use the Area to Find the Inner Radius (Rᵢ)
The area between the two circles can be expressed as:
\[ \text{Area} = \pi R₀^2 - \pi Rᵢ^2 \]
We know the area is 423.5 cm², so we can write:
\[ 423.5 = \pi (R₀^2 - Rᵢ^2) \]
Substituting \(R₀ = 21\) and \(\pi = \frac{22}{7}\):
\[ 423.5 = \frac{22}{7} (21^2 - Rᵢ^2) \]
Calculating \(21^2\):
\[ 21^2 = 441 \]
Now substituting this back into the equation:
\[ 423.5 = \frac{22}{7} (441 - Rᵢ^2) \]
To eliminate the fraction, multiply both sides by \(7\):
\[ 2964.5 = 22 (441 - Rᵢ^2) \]
Now divide both sides by \(22\):
\[ 134.75 = 441 - Rᵢ^2 \]
Rearranging gives:
\[ Rᵢ^2 = 441 - 134.75 \]
Calculating this:
\[ Rᵢ^2 = 306.25 \]

### Step 5: Find the Inner Radius (Rᵢ)
Now, take the square root to find \(Rᵢ\):
\[ Rᵢ = \sqrt{306.25} = 17.5 \text{ cm} \]

### Final Answer
The radius of the inner circle is \(7.5 \text{ cm}\).

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