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2015 AMC 12A Problem 7

Problem 7 of 25EasierAlgebraGeometry

Two right circular cylinders have the same volume. The radius of the second cylinder is 10%10\% more than the radius of the first. What is the relationship between the heights of the two cylinders?

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Solution

Let r,hr,h and R,HR,H be the radii and heights of the first and second cylinders. The volumes are equal, so πr2h=πR2H,\pi r^2 h = \pi R^2 H, and R=1.1r.R = 1.1r. Then πr2h=π(1.1r)2H\pi r^2 h = \pi(1.1r)^2 H =π(1.21r2)H.= \pi(1.21 r^2) H. Dividing by πr2\pi r^2 yields h=1.21H,h = 1.21H, so the first height is 21%21\% more than the second. Thus, the correct answer is D.

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Concepts: cylinder · volume · percentage

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