The area of a circular playground is \(22176\,\text{m}^2\). Find the cost of fencing this ground at the rate of Rs 50 per metre.
Rs 26,400
Step 1: Recall the formula for the area of a circle
The area of a circle is given by: \( A = \pi r^2 \)
Here, \( A = 22176\,\text{m}^2 \). We need to find the radius \( r \).
Step 2: Put the values into the formula
\( 22176 = \dfrac{22}{7} \times r^2 \)
Step 3: Solve for \( r^2 \)
\( r^2 = \dfrac{22176 \times 7}{22} \)
\( r^2 = 7056 \)
Step 4: Take the square root
\( r = \sqrt{7056} = 84\,\text{m} \)
So, the radius of the playground is 84 metres.
Step 5: Find the circumference (the boundary length)
Formula: \( C = 2\pi r \)
\( C = 2 \times \dfrac{22}{7} \times 84 \)
\( C = 528\,\text{m} \)
So, the fencing length needed is 528 metres.
Step 6: Find the cost of fencing
Rate of fencing = Rs 50 per metre
Total Cost = \( 528 \times 50 = 26,400 \)
Final Answer: Rs 26,400