Maths Olympiad Prep

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Geometry Difficulty 5.0 AIME Find the answer

Compute the side length of the largest cube contained in the region {(x,y,z):x2+y2+z2≤25 and x≥0}\{(x, y, z): x^{2}+y^{2}+z^{2} \leq 25 \text{ and } x \geq 0\} of three-dimensional space.

A number or a short expression. Spacing and $ signs are ignored.

Solution

The given region is a hemisphere, so the largest cube that can fit inside it has one face centered at the origin and the four vertices of the opposite face on the spherical surface. Let the side length of this cube be ss. Then, the radius of the circle is the hypotenuse of a triangle with side lengths ss and 22s\frac{\sqrt{2}}{2} s. So, by the Pythagorean Theorem, the radius equals 62s\frac{\sqrt{6}}{2} s. Since the radius of the hemisphere is 5, the side length of the cube is 563\frac{5 \sqrt{6}}{3}.

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