The ratio of the outer and inner perimeters of a circular path is 23:22. If the path is 5 m wide, what is the diameter of the inner circle ? Also , find the area of path enclosed between the two circle?

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the given ratio
The ratio of the outer perimeter (circumference) to the inner perimeter of the circular path is given as 23:22.

### Step 2: Set up the equations
Let the radius of the inner circle be \( r_1 \) and the radius of the outer circle be \( r_2 \). The relationship based on the given ratio can be expressed as:
\[ \frac{r_2}{r_1} = \frac{23}{22} \]
This implies:
\[ r_2 = \frac{23}{22} r_1 \]

### Step 3: Relate the radii to the width of the path
The width of the circular path is given as 5 m. This means:
\[ r_2 - r_1 = 5 \]

### Step 4: Substitute \( r_2 \) in terms of \( r_1 \)
Substituting the expression for \( r_2 \) from Step 2 into the equation from Step 3:
\[ \frac{23}{22} r_1 - r_1 = 5 \]

### Step 5: Solve for \( r_1 \)
Combine the terms:
\[ \left(\frac{23}{22} - 1\right) r_1 = 5 \]
\[ \left(\frac{23 - 22}{22}\right) r_1 = 5 \]
\[ \frac{1}{22} r_1 = 5 \]
Multiplying both sides by 22:
\[ r_1 = 5 \times 22 = 110 \text{ m} \]

### Step 6: Find the diameter of the inner circle
The diameter \( d_1 \) of the inner circle is given by:
\[ d_1 = 2 \times r_1 = 2 \times 110 = 220 \text{ m} \]

### Step 7: Find the area of the path enclosed between the two circles
The area of the path is the difference between the area of the outer circle and the inner circle:
\[ \text{Area} = \pi r_2^2 - \pi r_1^2 \]
Factoring out \( \pi \):
\[ \text{Area} = \pi (r_2^2 - r_1^2) \]

### Step 8: Calculate \( r_2 \)
Using the relationship \( r_2 = \frac{23}{22} r_1 \):
\[ r_2 = \frac{23}{22} \times 110 = 115 \text{ m} \]

### Step 9: Substitute \( r_2 \) and \( r_1 \) into the area formula
Now substitute \( r_1 \) and \( r_2 \):
\[ \text{Area} = \pi (115^2 - 110^2) \]
Calculating \( 115^2 \) and \( 110^2 \):
\[ 115^2 = 13225, \quad 110^2 = 12100 \]
Thus,
\[ \text{Area} = \pi (13225 - 12100) = \pi \times 1125 \]

### Step 10: Calculate the area using \( \pi \approx \frac{22}{7} \)
\[ \text{Area} = \frac{22}{7} \times 1125 = \frac{24750}{7} \approx 3535.71 \text{ m}^2 \]

### Final Answers:
- The diameter of the inner circle is **220 m**.
- The area of the path enclosed between the two circles is approximately **3535.71 m²**.

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