Problem:
Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are in diameter and high. Felicia buys cat food in cylindrical cans that are in diameter and high. What is the ratio of the volume of one of Alex's cans to the volume of one of Felicia's cans?
Answer Choices:
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Solution:
The volume of a cylinder with radius and height is . Alex's cans have radius and height , so the volume of one of his cans is = . Felicia's cans have radius and height , so the volume of one of her cans is = . Thus the ratio of the volume of one of Alex's cans to the volume of one of Felicia's cans is .
Because the volume of a cylindrical can depends upon the height and the square of the radius, when the height is doubled, the volume is multiplied by 2, and when the radius is divided by , the volume is divided by . Thus doubling the height and halving the radius results in Alex's cans having a volume that is one-half the volume of Felicia's cans. The requested ratio is therefore .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions