A race track is in the form of a ring whose outer and inner circumference are 506 m and 440 m respectively. Find the width of the track and also the area.

Answer

To solve the problem, we need to find the width of the race track and the area of the track. The race track is in the form of a ring, and we have the outer and inner circumferences given.

### Step-by-Step Solution:

**Step 1: Identify the given values.**
- Outer circumference (C₀) = 506 m
- Inner circumference (Cᵢ) = 440 m

**Step 2: Use the formula for circumference to find the outer radius (r₀).**
The formula for the circumference of a circle is:
\[ C = 2 \pi r \]
For the outer circumference:
\[ C₀ = 2 \pi r₀ \]
Substituting the value:
\[ 506 = 2 \times \frac{22}{7} \times r₀ \]

**Step 3: Solve for r₀.**
Rearranging the equation:
\[ r₀ = \frac{506 \times 7}{2 \times 22} \]
Calculating:
\[ r₀ = \frac{3542}{44} = 80.5 \, \text{m} \]

**Step 4: Use the formula for circumference to find the inner radius (rᵢ).**
For the inner circumference:
\[ Cᵢ = 2 \pi rᵢ \]
Substituting the value:
\[ 440 = 2 \times \frac{22}{7} \times rᵢ \]

**Step 5: Solve for rᵢ.**
Rearranging the equation:
\[ rᵢ = \frac{440 \times 7}{2 \times 22} \]
Calculating:
\[ rᵢ = \frac{3080}{44} = 70 \, \text{m} \]

**Step 6: Calculate the width of the track.**
The width of the track (W) is the difference between the outer radius and the inner radius:
\[ W = r₀ - rᵢ \]
Substituting the values:
\[ W = 80.5 - 70 = 10.5 \, \text{m} \]

**Step 7: Calculate the area of the track.**
The area (A) of the ring-shaped track can be calculated using the formula:
\[ A = \pi (r₀^2 - rᵢ^2) \]
Substituting the values:
\[ A = \frac{22}{7} \left( (80.5)^2 - (70)^2 \right) \]

**Step 8: Calculate the squares.**
Calculating \( (80.5)^2 \) and \( (70)^2 \):
\[ (80.5)^2 = 6480.25 \]
\[ (70)^2 = 4900 \]

**Step 9: Substitute back to find the area.**
\[ A = \frac{22}{7} (6480.25 - 4900) \]
\[ A = \frac{22}{7} \times 1580.25 \]
Calculating:
\[ A = \frac{22 \times 1580.25}{7} = 4966.5 \, \text{m}^2 \]

### Final Results:
- Width of the track = 10.5 m
- Area of the track = 4966.5 m²