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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