Cube or cuboid inscribed in a sphere: diameter = space diagonal

This type of question, asked 2 times: CDS 1 2018, CDS 2 2023. All 2 questions below, oldest first.

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Question 1
CDS 1 2018
86 $A cube of maximum volume (each corner touching the surface from inside) is cut from a sphere. What is the ratio of the volume of the cube to that of the sphere ?$
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Question 2
CDS 2 2023
51 $What is the radius of the sphere passing through the corners of the cuboid with edges 8 cm, 12 cm and 24 cm ?$
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