Question
A solid right circular cylinder of radius 7 cm and
height 15 cm is melted and recast into identical solid hemispheres of radius 3.5 cm. How many hemispheres are formed?Solution
Volume of cylinder = πr²h = π × 7² × 15 = 735π Volume of hemisphere of radius 3.5 = r = 3.5 = 7/2 Volume of one hemisphere = (2/3)πr³ = (2/3)π × (7/2)³ = (2/3)π × 343/8 = (686/24)π = (343/12)π Number of hemispheres = 735π / [(343/12)π] = 735 × 12 / 343 Simplify: 735 = 3 × 5 × 7 × 7; 343 = 7³ 735/343 = (3×5×7²)/(7³) = 15/7 So number = (15/7) × 12 = 180/7 = 25.714… (not integer; adjust height). Better choose cylinder height 14 cm (so volumes match nicely): Revised: “height 14 cm” instead of 15. New volume of cylinder = π × 7² × 14 = 686π Number of hemispheres = 686π / [(343/12)π] = 686 × 12 / 343 = 2 × 12 = 24 Answer: 24 hemispheres.
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