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A Curious Cone (Posted on 2004-10-11) Difficulty: 2 of 5
There is a cone whose radius is equal to its height. The numerical surface area is equal to the numerical volume. Find the exact radius and volume of the cone.

  Submitted by Jer    
Rating: 2.2500 (4 votes)
Solution: (Hide)
r = h = 3+3sqrt(2)
S.A. = V = (39+43sqrt(2))*pi
l = slant height = sqrt(r^2+h^2) = sqrt(2)*r
S.A. = pi*r^2 + pi*r*l
V = (1/3)*pi*r^2*h

Replace h with r, set S.A. = V and simplify to get r^3 - 6r^2 - 9 r = 0

Factor out r and solve via quadratic equation to get r.

Substitute r back into volume formula for V.

***As a geometry teacher I can verify that Surface Area is different from Lateral Area. Only the former counts the circular base.***

***As a geometry teacher I am embarassed at not pointing out that the cone is a right cone. I don't know of any S.A. formula for an oblique cone.***

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Some Thoughtsre: Open coneFletch2004-10-12 10:50:44
Open coneLarry2004-10-11 19:50:00
re: Seriously, though.....David Shin2004-10-11 15:33:26
Seriously, though.....Penny2004-10-11 14:55:03
re(3): More info required !Penny2004-10-11 14:11:06
Questionre(2): More info required !nikki2004-10-11 13:11:54
SolutionA solutionnikki2004-10-11 13:07:23
re: More info required !Penny2004-10-11 12:15:21
SolutionSolutionLarry2004-10-11 11:50:23
QuestionMore info required !Syzygy2004-10-11 11:48:43
SolutionSolution (I think...)Penny2004-10-11 10:11:10
SolutionFletch2004-10-11 09:57:57
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