Introduction / Context:
This is an annulus (ring) problem: the region between two concentric circles (outer edge of walkway and pool edge). We are given a ratio between the ring’s area and the pool’s area and must find the inner radius (pool radius).
Given Data / Assumptions:
- Pool radius = r (ft).
- Walkway width = 4 ft ⇒ outer radius = r + 4.
- Area of ring = π((r + 4)^2 − r^2) = 11/25 of πr^2 (pool area).
Concept / Approach:
Set up the area equation using ring area and simplify. Cancel π to reduce algebra. Solve the resulting quadratic for r and select the positive root.
Step-by-Step Solution:
π((r + 4)^2 − r^2) = (11/25) * πr^2(r + 4)^2 − r^2 = 11r^2 / 25r^2 + 8r + 16 − r^2 = 11r^2 / 25 ⇒ 8r + 16 = 11r^2 / 25Multiply by 25: 200r + 400 = 11r^2 ⇒ 11r^2 − 200r − 400 = 0Discriminant Δ = 200^2 + 4*11*400 = 57600 ⇒ √Δ = 240r = (200 + 240) / (2*11) = 440/22 = 20 (ft) (negative root discarded).
Verification / Alternative check:
Plug r = 20: Ring area = π((24)^2 − 20^2) = π(576 − 400) = 176π; Pool area = π * 400 = 400π; Ratio = 176/400 = 11/25.
Why Other Options Are Wrong:
10ft, 30ft, 40ft do not satisfy the precise ratio 11/25 when substituted.
Common Pitfalls:
Using outer diameter instead of outer radius in the ring formula.Forgetting to cancel π and mishandling algebra.
Final Answer:
20ft
Discussion & Comments