Problem 9 · 2019 AMC 8
Medium
Geometry & Measurement
volumevolume-scalingratio
Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are 6 cm in diameter and 12 cm high. Felicia buys cat food in cylindrical cans that are 12 cm in diameter and 6 cm high. What is the ratio of the volume of one of Alex's cans to the volume of one of Felicia's cans?
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Answer: B — 1:2.
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Hint 1 of 2
Don't plug into πr2h and crunch — a ratio only cares about how each dimension changes. Notice the diameter doubles and the height halves between the two cans.
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Hint 2 of 2
Radius appears squared, so doubling it multiplies volume by 4; halving the height multiplies by ½. The π cancels in any ratio — just combine the scale factors.
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Approach: scale factors, not actual volumes
- From Alex's can to Felicia's: the radius doubles and the height halves. In πr2h, radius is squared, so doubling it gives ×4; halving the height gives ×½.
- Felicia's volume = Alex's × 4 × ½ = Alex's × 2. So Alex : Felicia = 1 : 2.
- The trap: a length that's doubled changes area by 4 and volume by 8 — squared and cubed dimensions amplify. Track exponents when scaling, and let π cancel itself out in any ratio.
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