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.
Thanks to Steven Lord for catching the silly blunder in my other "solution". I hope this one is better.
Let the cone have radius 1 and height h. Its slant height = sqrt(1+h^2)
The hemisphere has radius r = h/sqrt(1+h^2).
cone surface area = pi*sqrt(1+h^2))+pi
hemisphere surface = 2*pi*r^2 = 2*pi*(h^2/(1+h^2))
Since both areas are in terms of h, we can write and solve the equation: 5*(cone) = 18*(hemisphere)
The becomes a biquadratic polygon with solution h^2=11/25
h=sqrt(11)/5
The vertex angle is 2*arctan(5/sqrt(11)) = 112.885 degrees
(I'm still puzzled by the 18/5 ratio chosen by the puzzle creator. It doesn't look arbitrary but nothing about the solution seems all that special.)
More of my solution is in this graph:
https://www.desmos.com/calculator/dta95twbrh
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Posted by Jer
on 2024-06-27 11:05:47 |
Corrected solution
