Radius of smaller circular park is 60 m and area of bigger circular park is `36 (1)/(9)% `more than area of smaller circular park. If side of a square park is half of the radius of bigger circular park and cost of fencing the square park is Rs.16 per m, then find the total cost of fencing the square park?

Options

A. A)2200 Rs.
B. B)2240 Rs.
C. C)2280 Rs.
D. D)2260 Rs.

Answer

To solve the problem step by step, we will follow the instructions provided in the video transcript.

### Step 1: Calculate the area of the smaller circular park.
The formula for the area of a circle is given by:
\[ \text{Area} = \pi r^2 \]
For the smaller circular park, the radius \( r \) is 60 m. Therefore, the area \( A_s \) of the smaller park is:
\[ A_s = \pi (60)^2 = \pi \times 3600 = 3600\pi \, \text{m}^2 \]

**Hint:** Remember that the area of a circle is calculated using the formula \( \pi r^2 \).

### Step 2: Calculate the area of the bigger circular park.
The area of the bigger circular park is 36 \( \frac{1}{9} \)% more than the area of the smaller park. First, convert \( 36 \frac{1}{9} \)% to a fraction:
\[ 36 \frac{1}{9} \% = \frac{36 \times 9 + 1}{9 \times 100} = \frac{325}{900} = \frac{13}{36} \]
This means the area of the bigger park \( A_b \) can be calculated as:
\[ A_b = A_s + \left(\frac{13}{36} \times A_s\right) = A_s \left(1 + \frac{13}{36}\right) = A_s \left(\frac{49}{36}\right) \]
Substituting \( A_s = 3600\pi \):
\[ A_b = 3600\pi \times \frac{49}{36} = 4900\pi \, \text{m}^2 \]

**Hint:** To find a percentage increase, convert the percentage to a fraction and multiply it by the original area.

### Step 3: Find the radius of the bigger circular park.
Using the area of the bigger park, we can find its radius \( r_b \):
\[ A_b = \pi r_b^2 \implies r_b^2 = \frac{A_b}{\pi} = \frac{4900\pi}{\pi} = 4900 \implies r_b = \sqrt{4900} = 70 \, \text{m} \]

**Hint:** To find the radius from the area, rearrange the area formula and take the square root.

### Step 4: Calculate the side of the square park.
The side \( a \) of the square park is half of the radius of the bigger circular park:
\[ a = \frac{r_b}{2} = \frac{70}{2} = 35 \, \text{m} \]

**Hint:** When the problem states "half of," simply divide the value by 2.

### Step 5: Calculate the perimeter of the square park.
The perimeter \( P \) of a square is given by:
\[ P = 4a = 4 \times 35 = 140 \, \text{m} \]

**Hint:** The perimeter of a square is four times the length of one side.

### Step 6: Calculate the total cost of fencing the square park.
The cost of fencing per meter is Rs. 16. Therefore, the total cost \( C \) is:
\[ C = \text{Cost per meter} \times \text{Perimeter} = 16 \times 140 = 2240 \, \text{Rs} \]

**Hint:** To find the total cost, multiply the cost per meter by the total length to be fenced.

### Final Answer:
The total cost of fencing the square park is Rs. 2240.

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