1. A circular swimming pool is surrounded by a concrete walk 8 feet wide. The area of the walk is 11/25 of the area of the pool. Find the radius of the pool
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Draw a diagram. You have two concentric circles. Let r denote the radius of the pool (in feet). Then the inner circle has radius r, and the outer circle has radius (r + 8). The area of the walk is the difference in area between that of the larger circle and that of the smaller circle:
A = π(r + 8)² - π r²
= π(r² + 16r + 64) - π r²
= 16π(r + 4)
This area is 11/25 of the area of the pool, which is π r². Therefore
16π(r + 4) = (11/25)(π r²)
16(r + 4) = (11/25)r²
400(r + 4) = 11r²
0 = 11r² - 400r - 1600
Using the quadratic formula, I get r = 40 or r = -40/11. The negative solution is unphysical, so we discard it. Thus, the radius of the pool is 40 ft.
A = π(r + 8)² - π r²
= π(r² + 16r + 64) - π r²
= 16π(r + 4)
This area is 11/25 of the area of the pool, which is π r². Therefore
16π(r + 4) = (11/25)(π r²)
16(r + 4) = (11/25)r²
400(r + 4) = 11r²
0 = 11r² - 400r - 1600
Using the quadratic formula, I get r = 40 or r = -40/11. The negative solution is unphysical, so we discard it. Thus, the radius of the pool is 40 ft.