Calculating the Outer Radius of a Circular Race Track

The question asks us to find the radius of the outer circle of a circular race track, given its inner circumference and width. We are given the inner circumference of the circular race track and its width.

First, we need to find the radius of the inner circle using the given inner circumference. The formula for the circumference of a circle is \(C = 2\pi r\), where \(C\) is the circumference and \(r\) is the radius.

Given:

• Inner Circumference (\(C_{inner}\)) = 440 cm
• Width of the track = 14 cm

We can find the inner radius (\(r_{inner}\)) using the inner circumference formula:

\[C_{inner} = 2\pi r_{inner}\]

Substituting the given value for \(C_{inner}\) and using the approximation \(\pi = \frac{22}{7}\):

\[440 = 2 \times \frac{22}{7} \times r_{inner}\]

\[440 = \frac{44}{7} \times r_{inner}\]

To find \(r_{inner}\), we rearrange the equation:

\[r_{inner} = \frac{440 \times 7}{44}\]

\[r_{inner} = \frac{10 \times 44 \times 7}{44}\]

\[r_{inner} = 10 \times 7\]

\[r_{inner} = 70 \text{ cm}\]

So, the radius of the inner circle is 70 cm.

The width of the race track is the difference between the radius of the outer circle and the radius of the inner circle.

Let \(r_{outer}\) be the radius of the outer circle.

Width of track = \(r_{outer} - r_{inner}\)

We are given that the width is 14 cm, and we just calculated \(r_{inner}\) as 70 cm.

\[14 = r_{outer} - 70\]

To find \(r_{outer}\), we add the inner radius to the width:

\[r_{outer} = r_{inner} + \text{Width of track}\]

\[r_{outer} = 70 \text{ cm} + 14 \text{ cm}\]

\[r_{outer} = 84 \text{ cm}\]

Therefore, the radius of the outer circle is 84 cm.

Let's quickly verify the calculations:

• Inner radius = 70 cm
• Outer radius = 84 cm
• Width = Outer radius - Inner radius = 84 cm - 70 cm = 14 cm (Matches the given width)
• Inner Circumference = \(2 \times \frac{22}{7} \times 70 = 2 \times 22 \times 10 = 440 \text{ cm}\) (Matches the given inner circumference)

The calculations are correct.

The final answer is 84 cm.