The ratio of the outer and inner perimeters of a circular patli is 23 : 22 If the path is 5 m wide, the diameter of the inner circle is

Options

A. 55m
B. 110 m
C. 220 m
D. 230 m

Explanation

To solve the problem, we need to find the diameter of the inner circle given the ratio of the outer and inner perimeters of a circular path and the width of the path. Here's a step-by-step breakdown of the solution:

### Step 1: Understand the Given Information
We know the ratio of the outer perimeter (circumference) to the inner perimeter is 23:22. The width of the path is given as 5 meters.

### Step 2: Define the Variables
Let:
- \( R_1 \) = inner radius of the circle
- \( R_2 \) = outer radius of the circle

Since the path is 5 meters wide, we can express the outer radius in terms of the inner radius:
\[ R_2 = R_1 + 5 \]

### Step 3: Set Up the Ratio of Perimeters
The perimeters (circumferences) of the circles can be expressed as:
- Inner perimeter = \( 2 \pi R_1 \)
- Outer perimeter = \( 2 \pi R_2 \)

Given the ratio of the outer to inner perimeters:
\[ \frac{2 \pi R_2}{2 \pi R_1} = \frac{23}{22} \]
This simplifies to:
\[ \frac{R_2}{R_1} = \frac{23}{22} \]

### Step 4: Substitute the Expression for \( R_2 \)
Substituting \( R_2 = R_1 + 5 \) into the ratio:
\[ \frac{R_1 + 5}{R_1} = \frac{23}{22} \]

### Step 5: Cross Multiply to Solve for \( R_1 \)
Cross multiplying gives us:
\[ 22(R_1 + 5) = 23R_1 \]
Expanding this:
\[ 22R_1 + 110 = 23R_1 \]

### Step 6: Rearrange the Equation
Rearranging the equation to isolate \( R_1 \):
\[ 110 = 23R_1 - 22R_1 \]
\[ 110 = R_1 \]

### Step 7: Calculate the Diameter of the Inner Circle
The diameter \( D \) of the inner circle is given by:
\[ D = 2R_1 = 2 \times 110 = 220 \text{ meters} \]

### Final Answer
The diameter of the inner circle is **220 meters**.

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