A circular park is surrounded by a road 21 m wide. If the park’s radius is 105 m, find the area of the road.
\(4851\pi\;\text{m}^2\) (≈ 15{,}226.1 m²)
Step 1: The park is a circle with radius \(105\,\text{m}\). This is the inner radius (r).
Step 2: The road is around the park and is \(21\,\text{m}\) wide. So, the outer radius (R) of the whole figure = \(105 + 21 = 126\,\text{m}\).
Step 3: Area of a circle is given by the formula \(A = \pi r^2\).
Step 4: Area of the big circle (park + road) = \(\pi R^2 = \pi (126^2)\).
Step 5: Area of the small circle (only the park) = \(\pi r^2 = \pi (105^2)\).
Step 6: The road area = (big circle area – small circle area).
So, \(\text{Area of road} = \pi (126^2 - 105^2)\).
Step 7: Use the identity \(a^2 - b^2 = (a-b)(a+b)\).
Here, \(a = 126\), \(b = 105\).
So, \(126^2 - 105^2 = (126 - 105)(126 + 105)\).
\(= 21 \times 231 = 4851\).
Step 8: Multiply with \(\pi\): Area = \(4851\pi\,\text{m}^2\).
Step 9: If we take \(\pi \approx 3.1416\), the value ≈ \(4851 \times 3.1416 = 15{,}226.1\,\text{m}^2\).
Final Answer: The area of the road is \(4851\pi\,\text{m}^2\) or about \(15{,}226.1\,\text{m}^2\) in SI units.