1954 AMC 12 Problem 27

Attempt Problem 27 of the 1954 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1954 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

27.

A right circular cone has for its base a circle having the same radius as a given sphere. The volume of the cone is one-half that of the sphere. The ratio of the altitude of the cone to the radius of its base is:
11\dfrac11
12\dfrac12
23\dfrac23
21\dfrac21
54\sqrt{\dfrac54}
Answer: D
Concepts:conespherevolume
Small hint:
Write both volumes using their common radius rr
Big hint:
Set 13πr2h\frac13\pi r^2h equal to half of 43πr3\frac43\pi r^3
Solution:
The volume condition is 13πr2h=12(43πr3). \frac13\pi r^2h=\frac12\left(\frac43\pi r^3\right). Cancelling the common factors gives h=2r,h=2r, so the requested ratio is 2:1.2:1.
Thus, the correct answer is D.
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