A circular park with 8 m diameter is surrounded by 0.5 m wide path. What is the percentage increase in area?

Correct Option / Answer:
The correct answer is 26.56%.

Step-by-Step Explanation:

Step 1: Understand the given dimensions
We are given a circular park with a diameter of 8 m.
The radius of the inner circular park, r, is:
$r=\frac{8\text{m}}{2}=4\text{m}$

A path of width 0.5 m surrounds the park. This forms an outer circle.
The radius of the outer circle including the path, R, is:
$R=r+\text{width of path}=4\text{m}+0.5\text{m}=4.5\text{m}$

Step 2: Calculate the original area of the circular park
The area of a circle is given by the formula $A=\pi r^2$.
Original Area ($A_{\text{original}}$):
$A_{\text{original}}=\pi \times 4^2=16\pi {\text{m}}^2$

Step 3: Calculate the total area including the path
Total Area ($A_{\text{total}}$):
$A_{\text{total}}=\pi \times {4.5}^2=20.25\pi {\text{m}}^2$

Step 4: Calculate the increase in area
The increase in area is the difference between the total area and the original area:
$\text{Increase in Area}=A_{\text{total}}-A_{\text{original}}=20.25\pi -16\pi =4.25\pi {\text{m}}^2$

Step 5: Calculate the percentage increase in area
The percentage increase in area is calculated as:
$\text{Percentage Increase}=\left(\frac{\text{Increase in Area}}{A_{\text{original}}}\right)\times 100$

Substituting the values:
$\text{Percentage Increase}=\left(\frac{4.25\pi }{16\pi }\right)\times 100$

$\text{Percentage Increase}=\frac{4.25}{16}\times 100=\frac{425}{16}=26.5625\%$

Rounding off to two decimal places gives 26.56%.