A right circular cone has base of radius 1 and height 3.
A cube is inscribed in the cone
so that one face of the cube is contained in the base of the cone.
What is the side's length of
the cube?
Source: Putnam 1998
On a plane perpendicular to the base containing a diagonal of cube's base square, a vertical edge of the square forms one leg of a right triangle where the hypotenuse lies along an apothem of the cone.
The angle opposite the sought edge is arctan(3). The other leg or the right triangle is 1 minus half the diagonal length of the face of the cube.
If the side length sought is s, then
s = 3 * (1 - s*sqrt(2)/2)
where the 3 is tan(arctan(3)).
Wolfram Alpha solves as
s = 6/(2 + 3 sqrt(2)) =~ 0.961131723051122.
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Posted by Charlie
on 2024-04-10 09:18:54 |
solution
