A sphere of radius r cm is packed in a box of cubical shape. What should be the minimum volume (in cm³) of the box that can enclose the sphere?
Options:
(A) r³/8
(B) r³
(C) 2r³
(D) 8r³
Official Answer Key:
D
Step-by-Step Solution & Explanation
• For a sphere of radius r to just fit inside a cube (minimum enclosing cube), the cube's side length must equal the sphere's diameter, since the sphere touches all six faces of the cube.
• Side of cube, a = diameter of sphere = 2r
• Minimum volume of cube = a³ = (2r)³ = 8r³
• Distractor logic: (A) uses a = r/2. (B) uses a = r (physically impossible). (C) applies a linear scaling error instead of cubing.
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