Problem 14 · 2011 Math Kangaroo
Stretch
Geometry & Measurement
difference-of-squares
I have two cubes with side lengths a dm and a + 1 dm. The big cube is full of water and the little one is empty. I pour as much water as possible from the big one into the little one, and now 217 ℓ remain in the big cube. How many litres of water are now in the little one?
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Answer: B — 512 ℓ
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Hint 1 of 2
The leftover water equals the big cube's volume minus the small cube's volume.
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Hint 2 of 2
Set that difference equal to 217 (in litres = dm³) and solve for the side a.
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Approach: solve the cube-volume difference for a
- 1 dm³ = 1 litre, so the leftover is (a+1)³ − a³ = 217.
- Expanding: 3a² + 3a + 1 = 217, so a² + a − 72 = 0, giving a = 8.
- The small cube is now full, holding a³ = 8³ = 512 litres.
- So 512 ℓ, choice (B).
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