Problem 30 · 2022 Math Kangaroo
Stretch
Geometry & Measurement
pythagorean-triplespatial-reasoning
A hemispheric hole is carved into each face of a wooden cube with sides of length 2. All holes are equally sized, and their midpoints are in the centre of the faces of the cube. The holes are as big as possible so that each hemisphere touches each adjacent hemisphere in exactly one point. How big is the diameter of the holes?
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Answer: C — \(\sqrt{2}\)
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Hint 1 of 2
Adjacent hemispheres touch along an edge of the cube, where their rims meet.
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Hint 2 of 2
Find the distance between the centres of two adjacent faces and set 2r equal to it.
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Approach: set the diameter equal to the distance between adjacent face centres
- For a cube of side 2, the centres of two adjacent faces are sqrt(1^2 + 1^2) = sqrt(2) apart.
- Hemispheres on those faces just touch when their radii meet along that line: r + r = sqrt(2).
- So the diameter 2r = sqrt(2).
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