# Cubes and spheres... Spheres and cubes... Luke's gone mad.

Calculus Level pending

Luke inscribed a sphere in a cube of sides $$x$$. Then he inscribed another cube in the sphere. Assume he could do this process infinitely. Let the sum of the volumes of all the spheres be $$V_{s_{1}}$$, the sum of the volumes of all the cubes be $$V_{c_{1}}$$ and the quotient $$\dfrac{V_{s_{1}}}{V_{c_{1}}}$$ be $$Q_{1}$$. If Luke started the process by a sphere of radius $$x$$ instead of a cube of sides $$x$$, the sum of the volumes of all the spheres would be $$V_{s_{2}}$$ and the sum of the volumes of all the cubes would be $${V_{c_{2}}}$$. Let $$\dfrac{V_{s_{2}}}{V_{c_{2}}}$$ be $$Q_{2}$$. If the quotient $$\dfrac{Q_{1}}{Q_{2}}$$ can be written as $$k^{-\tfrac{3}{2}}$$, find $$k$$.

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