A square park has side 100 m (area = 10,000 m^2). The area outside a central circular lawn but inside the square is 8,614 m^2. Find the radius of the circular lawn.

Introduction / Context:
The problem gives the square’s total area and the square area remaining after removing a centrally placed circular lawn. The difference is the circle’s area. From that, compute the radius by applying the circle area formula. The values are chosen so π = 22/7 yields perfect squares for r^2.

Given Data / Assumptions:

• Square side s = 100 m ⇒ area A_square = 10,000 m^2.
• Area outside circle but inside square = 8,614 m^2.
• Hence circle area A_circ = 10,000 − 8,614 = 1,386 m^2.
• Use π = 22/7 unless noted otherwise.

Concept / Approach:
Area of a circle is A_circ = πr^2. Thus r^2 = A_circ / π. With π = 22/7, division becomes multiplication by 7/22, often leading to a neat square for r^2 if the problem is crafted cleanly.

Step-by-Step Solution:
A_circ = 1,386 m^2r^2 = 1,386 * (7/22) = 441r = √441 = 21 m

Verification / Alternative check:
Back-calc: πr^2 = (22/7)*441 = 22*63 = 1,386 m^2, consistent. Subtracting from 10,000 gives 8,614 m^2 as stated.

Why Other Options Are Wrong:
31 m and 41 m produce circle areas that do not equal 1,386 m^2 with π = 22/7; only 21 m satisfies the equality.

Common Pitfalls:
Forgetting to convert the complement area to the circle area, or using π ≈ 3.14 and rounding prematurely. With π = 22/7 the arithmetic is exact.

Final Answer:
21 m