Geometry and TrigonometryVery hardEmpty space around a sphere in a cube
Geometry and Trigonometry practice question
A sphere of radius 3 fits exactly inside a cube, touching all six faces. The volume of the cube not occupied by the sphere can be written as a − bπ, where a and b are constants. What is the value of a + b?
Answer: 252
The cube has edge 6 and volume 216; the sphere's volume is (4/3)π(27) = 36π. So a = 216, b = 36, and a + b = 252.
Common wrong answers
- If you answered 180
- 180 subtracts 36 from 216, mixing the two constants.
- If you answered 228
- 228 uses b = 12, a slip in the sphere's volume.
- If you answered 243
- 243 uses a sphere volume of 27π, forgetting the 4/3.
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