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How to solve this problem:

Contest maths problem

Also, most people would use trigonometry, but is there a way to use derivative to solve this too?

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up vote 7 down vote accepted

enter image description here Above, $O$ is the centre of the sphere, $x$ is its radius. Writing the area of trapezoid $ABCD$ as a sum of three triangles, we can easily solve for $x$ to get:

$$ x=\frac{1200\sqrt{3}}{54}=38.490017\ldots $$

No trig or calculus needed, only geometry.

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R - radius of the cone, h - height of the cone. Point A is the foot of the perpendicular from the centre of the sphere to the base. It is easy to calculate the distance from A to the centre of a base circle to be $2R\sqrt{3}/3$. –  Theo Nov 4 '12 at 19:51
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