The area of circular garden is `5544m^(2)` .Outside this garden , a path of uniform width is laid all around . The area of the path is `2002m^(2)` .Find :
(i) the radius of the circular garden .
(ii) the width of the path.

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Calculate the radius of the circular garden

The area of the circular garden is given as \( 5544 \, m^2 \). The formula for the area of a circle is:

\[ \text{Area} = \pi r^2 \]

where \( r \) is the radius. We can rearrange this formula to find \( r^2 \):

\[ r^2 = \frac{\text{Area}}{\pi} \]

Substituting the given area:

\[ r^2 = \frac{5544}{\pi} \]

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

\[ r^2 = \frac{5544 \times 7}{22} \]

Calculating the numerator:

\[ 5544 \times 7 = 38808 \]

Now, divide by 22:

\[ r^2 = \frac{38808}{22} = 1764 \]

Now, take the square root to find \( r \):

\[ r = \sqrt{1764} = 42 \, m \]

### Step 2: Calculate the area of the outer circle

The area of the path is given as \( 2002 \, m^2 \). The total area of the outer circle (garden + path) can be calculated as:

\[ \text{Area of outer circle} = \text{Area of garden} + \text{Area of path} \]

Substituting the values:

\[ \text{Area of outer circle} = 5544 + 2002 = 7546 \, m^2 \]

### Step 3: Calculate the radius of the outer circle

Using the area of the outer circle, we can find its radius \( R \):

\[ R^2 = \frac{\text{Area of outer circle}}{\pi} \]

Substituting the total area:

\[ R^2 = \frac{7546}{\pi} \]

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

\[ R^2 = \frac{7546 \times 7}{22} \]

Calculating the numerator:

\[ 7546 \times 7 = 52822 \]

Now, divide by 22:

\[ R^2 = \frac{52822}{22} = 2401 \]

Now, take the square root to find \( R \):

\[ R = \sqrt{2401} = 49 \, m \]

### Step 4: Calculate the width of the path

The width of the path can be found by subtracting the radius of the garden from the radius of the outer circle:

\[ \text{Width of the path} = R - r = 49 - 42 = 7 \, m \]

### Final Answers:

(i) The radius of the circular garden is \( 42 \, m \).

(ii) The width of the path is \( 7 \, m \).

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