Calculating the Cost of a Circular Garden Path

The problem asks us to find the cost of constructing a path inside a circular garden. We are given the radius of the garden, the width of the path, the rate per square meter for making the path, and the value of \(\pi\) to use.

Understanding the Geometry of the Circular Path

The garden is circular with a radius of 15 m. A path 2 m wide is to be made inside the garden. This means the path forms an annular region (a ring) between two concentric circles.

• The outer circle is the boundary of the garden itself. Its radius is given as \(R = 15\) m.
• The inner circle is the boundary of the garden *excluding* the path. Since the path is 2 m wide and is inside, the radius of the inner circle is the radius of the garden minus the width of the path. So, the inner radius is \(r = 15 - 2 = 13\) m.

Calculating the Area of the Circular Path

To find the cost, we first need to calculate the area of the path. The area of the path is the area of the outer circle minus the area of the inner circle.

The area of a circle is given by the formula \(A = \pi \times (\text{radius})^2\).

• Area of the outer circle (\(A_{outer}\)) = \(\pi R^2 = \pi \times (15)^2\)
• Area of the inner circle (\(A_{inner}\)) = \(\pi r^2 = \pi \times (13)^2\)

The area of the path (\(A_{path}\)) is \(A_{outer} - A_{inner}\).

\[A_{path} = \pi R^2 - \pi r^2\]

We can factor out \(\pi\):

\[A_{path} = \pi (R^2 - r^2)\]

Now, substitute the values \(R=15\) m, \(r=13\) m, and \(\pi = \frac{22}{7}\):

\[A_{path} = \frac{22}{7} (15^2 - 13^2)\]

Calculate the squares:

\[A_{path} = \frac{22}{7} (225 - 169)\]

Subtract the values inside the parenthesis:

\[A_{path} = \frac{22}{7} (56)\]

Now, perform the multiplication. Since 56 is divisible by 7 (\(56 \div 7 = 8\)), we get:

\[A_{path} = 22 \times 8\]

\[A_{path} = 176 \text{ sq. m.}\]

The area of the circular path is 176 square meters.

Calculating the Total Cost of Making the Path

The rate for making the path is given as ₹ 24 per square meter.

The total cost is the area of the path multiplied by the rate per square meter.

Total Cost = Area of path \(\times\) Rate per sq. m

Total Cost = \(176 \text{ sq. m} \times ₹ 24 \text{/sq. m}\)

\[\text{Total Cost} = 176 \times 24\]

Let's calculate \(176 \times 24\):

The total cost of making the path is ₹ 4224.

Conclusion

The cost of making the 2 m wide path inside the circular garden of radius 15 m at the rate of ₹ 24 per sq. m is ₹ 4224.

Revision Table: Circular Path Calculation

Additional Information: Area of Annulus

An annulus is the region between two concentric circles. In this problem, the circular path is an annulus. 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:

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

This formula was used directly in our calculation. The term \(R^2 - r^2\) can also be factored as \((R-r)(R+r)\), which gives another way to express the area:

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

In our case, \(R=15\) and \(r=13\):

• \(R-r = 15 - 13 = 2\) (which is the width of the path)
• \(R+r = 15 + 13 = 28\)

So, the area is \(\pi \times 2 \times 28 = 56\pi\).

Using \(\pi = \frac{22}{7}\):

\[\text{Area} = \frac{22}{7} \times 56 = 22 \times 8 = 176 \text{ sq. m}\]

This confirms the previous calculation method and shows an alternative way to calculate the area of the annular path.