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Square in a Cone (Posted on 2016-07-29) Difficulty: 3 of 5
Consider a hollow right circular cone having radius 20 and height 16.

Find the maximum possible area of the square that can lie inside the cone.

No Solution Yet Submitted by K Sengupta    
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Some Thoughts Thoughts Comment 2 of 2 |
Any square lies on a plane.  When that plane slices the cone, the intersection is an ellipse, call it E.

Now slice a second plane through the cone perpendicular to the cone's axis and containing the center of the ellipse.  That plane intersects the cone in a circle, call it C.

The diameter of C is equal to the minor axis of E.  The inscribed square in E is larger than the inscribed square in C.  But in order to have a chance to be larger than the square inscribed in the base, the major axis of E must be longer than the diameter of the base.

But increasing the length of the major axis of E requires tilting the slicing plane increasingly steeper against the base, which has the effect of shrinking the minor axis.

So based on this handwaving argument I conjecture that the largest square is the square inscribed in the base, regardless of the values chosen for the radius and height.

  Posted by Brian Smith on 2016-07-30 11:00:40
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