Problem 13 · 2019 Math Kangaroo
Hard
Geometry & Measurement
areadivision
A cuboid-shaped container that is not completely filled holds 120 m³ of water. Depending on which face the container stands on, the depth of the water is 2 m, 3 m or 5 m (drawings not to scale). What is the volume of the container?
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Answer: E — 240 m³
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Hint 1 of 2
The same 120 m³ gives depth 2, 3 or 5 on three different bottom faces, so each face area is 120 divided by that depth.
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Hint 2 of 2
From the three face areas 60, 40 and 24, the cuboid's volume is the square root of their product.
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Approach: the water volume is the same in every orientation
- The 120 m³ of water sits with depth 2, 3 or 5 m on three different faces.
- Each bottom face area = 120 ÷ depth, giving 60, 40 and 24 m² — the three face areas of the cuboid.
- If the edges are x, y, z then the faces are xy, xz, yz, so \((xyz)^2 = 60\cdot 40\cdot 24 = 57600\) and the volume is \(xyz = 240\).
- Answer (E) 240 m³.
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