A hemisphere is inscribed in a cone so that its base lies on the base of the cone. The ratio of the area of the entire surface of the cone to the area of the hemisphere (without the base) is 18/5. Compute the angle at the vertex of the cone.
EDIT *** This is incorrect. The cone and hemisphere do not have the same radius. That's why it looked too easy.***
This solution does not make it look like the problem is D4. I hope it is correct.
entire cone = pi*r^2+pi*r*l (l is the slant height)
hemisphere = 2pi*r^2
reduce this ratio and cross mulitily with 18/5:
5r + 5l = 36r
5l = 31r
The vertex angle, theta, is twice the angle formed by the axis of the cone with the slant height.
sin(theta/2) = 5/31
theta = 18.56 degrees
I assumed the cone is a right cone. I don't think a hemisphere can be inscribed in an oblique cone. I also don't know if an oblique cone has a single vertex angle.
Edited on June 27, 2024, 10:22 am
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Posted by Jer
on 2024-06-25 15:36:05 |
Solution
