Understanding the Circular Park Area Problem

The question asks us to find the area of a path that surrounds a circular park. We are given the area of the circular park and the width of the path. To solve this, we need to first find the radius of the park using its area. Then, we can find the radius of the larger circle formed by the park plus the path. Finally, the area of the path will be the difference between the area of the larger circle and the area of the smaller circle (the park).

We are given:

• Area of the circular park = \(12474 \text{ m}^2\)
• Width of the path = \(3.5 \text{ m}\)
• Value of \(\pi\) to use = \(\frac{22}{7}\)

Calculating the Radius of the Circular Park

The area of a circle is given by the formula \(\text{Area} = \pi r^2\), where \(r\) is the radius of the circle.

We know the area of the park is \(12474 \text{ m}^2\). Let \(r_p\) be the radius of the park.

So, \(12474 = \pi r_p^2\)

Substitute the value of \(\pi = \frac{22}{7}\):

\(12474 = \frac{22}{7} \times r_p^2\)

Now, we solve for \(r_p^2\):

\(r_p^2 = \frac{12474 \times 7}{22}\)

\(r_p^2 = \frac{87318}{22}\)

Let's perform the division:

\(r_p^2 = 3969\)

To find the radius \(r_p\), we take the square root of 3969:

\(r_p = \sqrt{3969}\)

\(r_p = 63 \text{ m}\)

So, the radius of the circular park is \(63 \text{ m}\).

Calculating the Radius of the Park and Path Together

The path has a width of \(3.5 \text{ m}\) and is around the park. This means the outer radius (radius of the park plus the path) is the radius of the park plus the width of the path.

Let \(R\) be the radius of the larger circle (park + path).

\(R = \text{Radius of park} + \text{Width of path}\)

\(R = r_p + 3.5\)

\(R = 63 + 3.5\)

\(R = 66.5 \text{ m}\)

The radius of the larger circle is \(66.5 \text{ m}\).

Calculating the Area of the Path Around the Circular Park

The area of the path is the area of the larger circle (park + path) minus the area of the smaller circle (park).

\(\text{Area of path} = \text{Area of larger circle} - \text{Area of park}\)

Area of larger circle = \(\pi R^2\)

\(\text{Area of path} = \pi R^2 - \pi r_p^2\)

\(\text{Area of path} = \pi (R^2 - r_p^2)\)

Substitute the values \(R = 66.5\), \(r_p = 63\), and \(\pi = \frac{22}{7}\):

\(\text{Area of path} = \frac{22}{7} ((66.5)^2 - (63)^2)\)

First, calculate the squares:

\((66.5)^2 = 66.5 \times 66.5 = 4422.25\)

\((63)^2 = 63 \times 63 = 3969\)

Now, calculate the difference:

\((66.5)^2 - (63)^2 = 4422.25 - 3969\)

\(4422.25 - 3969 = 453.25\)

Now, substitute this back into the area of path formula:

\(\text{Area of path} = \frac{22}{7} \times 453.25\)

Multiply 22 by 453.25:

\(22 \times 453.25 = 9971.5\)

Now, divide by 7:

\(\text{Area of path} = \frac{9971.5}{7}\)

\(\frac{9971.5}{7} = 1424.5\)

So, the area of the path is \(1424.5 \text{ m}^2\).

Summary of Circular Area Calculations

Here is a summary of the key values calculated:

The final answer for the area of the path around the circular park is \(1424.5 \text{ m}^2\).

Revision Table: Circular Area Calculations

Additional Information: Areas of Rings

The area of a path around a circular park is an example of finding the area of an annulus or a circular ring. An annulus is the region between two concentric circles. If the radius of the outer circle is \(R\) and the radius of the inner circle is \(r\), the area of the annulus is given by the formula:

\(\text{Area of Annulus} = \pi R^2 - \pi r^2\)

This can also be factored as:

\(\text{Area of Annulus} = \pi (R^2 - r^2)\)

Using the difference of squares formula (\(a^2 - b^2 = (a-b)(a+b)\)), we can also write it as:

\(\text{Area of Annulus} = \pi (R - r)(R + r)\)

In our problem, \(R - r\) is the width of the path, which is \(3.5 \text{ m}\). \(R+r\) is the sum of the radii. So, the area of the path can also be calculated as \(\pi \times 3.5 \times (66.5 + 63) = \frac{22}{7} \times 3.5 \times 129.5\).

\(\frac{22}{7} \times 3.5 = 22 \times 0.5 = 11\)

Area of path = \(11 \times 129.5\)

\(11 \times 129.5 = 1424.5\)

This confirms our previous calculation and shows an alternative way to calculate the area of the circular path using the radii and the path width.