A circular swimming pool is surrounded by a concrete wall that is 4 ft wide (uniform width).\nThe area of the concrete wall (the ring between the outer circle and inner pool) is 11/25 of the area of the pool.\nFind the radius of the pool in feet.

Introduction:
This question tests area relationships in concentric circles (a ring or annulus). The pool is the inner circle with radius r. The concrete wall creates an outer circle with radius (r + 4). The wall area is the difference between outer area and inner area. The problem gives a fractional relationship between wall area and pool area, which leads to a quadratic equation in r. Solving that quadratic carefully yields the pool’s radius. This is a multi-step geometry and algebra integration question, making it harder than direct formula substitution.

Given Data / Assumptions:

• Pool radius = r ft
• Wall width = 4 ft
• Outer radius = r + 4
• Pool area = pi * r^2
• Wall area = pi * (r + 4)^2 - pi * r^2
• Given: wall area = (11/25) * pool area

Concept / Approach:
Set up: pi((r+4)^2 - r^2) = (11/25)*pi*r^2. Cancel pi to simplify. Expand (r+4)^2 - r^2 to get a linear expression in r, then solve the resulting quadratic equation for r. Choose the positive root because radius must be positive.

Step-by-Step Solution:
Wall area: (r+4)^2 - r^2 = (r^2 + 8r + 16) - r^2 = 8r + 16Equation: 8r + 16 = (11/25) * r^2Multiply by 25: 200r + 400 = 11r^2Rearrange: 11r^2 - 200r - 400 = 0Discriminant = 200^2 + 4*11*400 = 40000 + 17600 = 57600sqrt(57600) = 240r = (200 + 240) / (2*11) = 440/22 = 20

Verification / Alternative Check:
With r = 20, outer radius = 24. Pool area = pi*400. Wall area = pi*(576 - 400) = pi*176. Ratio wall/pool = 176/400 = 11/25, exactly matches the condition. This confirms r = 20 ft is correct and also shows why the other root (negative) is invalid for a radius.

Why Other Options Are Wrong:
10, 30, 40, 16 ft: these do not satisfy the exact fractional area condition 11/25 when substituted into the ring-area ratio.

Common Pitfalls:
Forgetting to subtract inner area from outer area for wall area.Not cancelling pi, leading to messy calculations.Expanding (r+4)^2 incorrectly.Choosing the negative quadratic root, which is not physically meaningful.

Final Answer:
20 ft