The radii of the inner and outer circumferences of a circular-running-track are 63 m and 70 m respectively. Find :
(i) the area of the track
(ii) the difference between the lengths of the two circumferences of the track

Answer

To solve the problem, we need to find two things:

1. The area of the circular track.
2. The difference between the lengths of the two circumferences of the track.

Let's break down the solution step by step.

### Step 1: Identify the radii
The radii of the inner and outer circumferences of the circular track are given as:
- Inner radius (R2) = 63 m
- Outer radius (R1) = 70 m

### Step 2: Calculate the area of the track
The area of the track can be found by subtracting the area of the inner circle from the area of the outer circle.

The formula for the area of a circle is:
\[ \text{Area} = \pi r^2 \]

Thus, the area of the outer circle (C1) is:
\[ \text{Area of C1} = \pi R1^2 = \pi (70)^2 \]

And the area of the inner circle (C2) is:
\[ \text{Area of C2} = \pi R2^2 = \pi (63)^2 \]

Now, the area of the track is:
\[ \text{Area of the track} = \text{Area of C1} - \text{Area of C2} \]
\[ = \pi (70^2) - \pi (63^2) \]
\[ = \pi (70^2 - 63^2) \]

### Step 3: Calculate \(70^2\) and \(63^2\)
Calculating the squares:
- \(70^2 = 4900\)
- \(63^2 = 3969\)

Now, substitute back into the area equation:
\[ \text{Area of the track} = \pi (4900 - 3969) \]
\[ = \pi (931) \]

### Step 4: Substitute the value of \(\pi\)
Using \(\pi \approx \frac{22}{7}\):
\[ \text{Area of the track} = \frac{22}{7} \times 931 \]

### Step 5: Calculate the area
Calculating:
\[ \text{Area of the track} = \frac{22 \times 931}{7} \]
\[ = \frac{20482}{7} \]
\[ = 2926 \text{ m}^2 \]

### Step 6: Calculate the difference between the lengths of the two circumferences
The formula for the circumference of a circle is:
\[ \text{Circumference} = 2 \pi r \]

Thus, the circumference of the outer circle (C1) is:
\[ \text{Circumference of C1} = 2 \pi R1 = 2 \pi (70) \]

And the circumference of the inner circle (C2) is:
\[ \text{Circumference of C2} = 2 \pi R2 = 2 \pi (63) \]

The difference between the two circumferences is:
\[ \text{Difference} = \text{Circumference of C1} - \text{Circumference of C2} \]
\[ = 2 \pi (70) - 2 \pi (63) \]
\[ = 2 \pi (70 - 63) \]
\[ = 2 \pi (7) \]

### Step 7: Substitute the value of \(\pi\) again
\[ \text{Difference} = 2 \times \frac{22}{7} \times 7 \]

### Step 8: Calculate the difference
The \(7\) cancels out:
\[ \text{Difference} = 2 \times 22 = 44 \text{ m} \]

### Final Answers
1. The area of the track is \(2926 \text{ m}^2\).
2. The difference between the lengths of the two circumferences is \(44 \text{ m}\).