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2021 AMC 12B Problem 6

Problem 6 of 25EasierGeometry

An inverted cone with base radius 1212 cm and height 1818 cm is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of 2424 cm. What is the height in centimeters of the water in the cylinder?

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Solution

The cone holds 13π(12)2(18)=864π\dfrac13\pi(12)^2(18)=864\pi cubic centimeters of water. Poured into the cylinder, this fills to height hh where π(24)2h=864π.\pi(24)^2 h=864\pi. Then 576h=864,576h=864, so h=1.5.h=1.5. Thus, the correct answer is A.

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Concepts: cone · cylinder · volume

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.