SOLUTION: A cone of height 12cm has a volume of n cm cubed and a curved surface area of n cm squared. Find the vertical angle of the cone.

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Question 1156779: A cone of height 12cm has a volume of n cm cubed and a curved surface area of n cm squared. Find the vertical angle of the cone.
Answer by greenestamps(13198) About Me  (Show Source):
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V+=+%281%2F3%29%28pi%29%28r%5E2%29%28h%29

A+=+%28pi%29%28r%29%28l%29 where l is the slant height.

The volume and the lateral surface area are numerically equal:

%281%2F3%29%28pi%29%28r%5E2%29%2812%29+=+%28pi%29%28r%29%28l%29
4r+=+l

The radius of the cone r and the slant height 4r form a right triangle with the altitude of the cone. The angle in that triangle at the vertex of the cone is half the vertical angle of the cone; the sine of that angle is r/(4r) = 1/4.

The vertical angle of the cone is

2%2Asin%5E%28-1%29%281%2F4%29 = approximately 28.955 degrees.