Nadia glued some 1-cm wooden cubes together to form a cuboid. She then dipped the whole cuboid into a tin of blue varnish, so the whole outside was coated. Exactly 30 of the small cubes had no varnish on any of their faces.
- What was the smallest possible volume of the cuboid? (2 marks)
- For the cuboid in (a), how many of the small cubes had varnish on at least one face? (1 mark)
Need a hint?
The 30 cubes with no varnish make a smaller cuboid hidden in the middle. Think about which three numbers multiply to 30, and how much longer each side of the whole cuboid is.
Worked answer
- (a) The cubes with no varnish form a smaller cuboid inside. It has 30 cubes, so its length × width × height = 30.
- Each side of the whole cuboid is 2 cm longer than the inside block, because there is one layer of varnished cubes on each end.
- To get the smallest whole cuboid, the inside block should be as close to a cube shape as possible: 2 × 3 × 5 = 30.
- The whole cuboid is (2 + 2) × (3 + 2) × (5 + 2) = 4 × 5 × 7 = 140 cubes.
- Other inside blocks give larger volumes: 1 × 5 × 6 gives 3 × 7 × 8 = 168, and 1 × 3 × 10 gives 3 × 5 × 12 = 180.
- So the smallest possible volume is 140 cm³.
- (b) Cubes with varnish = all cubes − cubes with no varnish = 140 − 30 = 110.
The mistake this question tests
Factors and multiples (HCF, LCM)
Many pupils add only 1 to each side of the inside block instead of 2, forgetting that there is a layer of varnished cubes on both ends of every side. Others pick 1 × 1 × 30 without checking whether another set of factors gives a smaller cuboid.
How to fix it at home →More PSLE area & volume questions
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See the answer →Short answer · 2mNoah's craft box has a square base with an area of 49 cm²
See the answer →Structured · 5mThree identical rectangles are placed side by side to form one large rectangle
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See the answer →Practise PSLE area & volume until the mistakes stop.
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