SOLUTION: A cone of height 9cm has a volume of ncm^3 and curved surface area of ncm^2. Find the vertical angle

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Question 1136494: A cone of height 9cm has a volume of ncm^3 and curved surface area of ncm^2. Find the vertical angle
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Let r be the radius of the cone. With height 9, the slant height l is then sqrt%28r%5E2%2B81%29.

The problem says that, ignoring units, the volume and lateral surface area are the same. So

%281%2F3%29%28pi%29%28r%5E2%29%28h%29+=+%28pi%29%28r%29%28l%29
%281%2F3%29%28pi%29%28r%5E2%29%289%29+=+%28pi%29%28r%29%28sqrt%28r%5E2%2B81%29%29
3r+=+sqrt%28r%5E2%2B81%29
9r%5E2+=+r%5E2%2B81

I'm not sure of what your definition of "vertical angle" is in this problem. The angle between the base and the slant height of the cone is

arccos%28r%2Fsqrt%28r%5E2%2B81%29%29+=+arccos%28r%2F%283r%29%29+=+arccos%281%2F3%29

which is 70.53 degrees, to 2 decimal places.