A rectangular wooden block is 6 cm long and 4 cm wide, and its height is h cm. A cube is cut out from one corner of the block. The volume of the cube removed is 8 cm³.
- Find the length of an edge of the cube removed.
- The volume of the remaining wooden block is 88 cm³. Find the height h of the original block.
- A solid cube of side 4 cm is painted on all its faces and then cut into 1-cm cubes. How many of the 1-cm cubes have exactly 2 faces painted?
Need a hint?
(a) Think of a number that multiplies by itself three times to give 8. (b) Add back the removed volume first. (c) Corner cubes have 3 painted faces, so leave them out of the count for each edge.
Worked answer
- (a) Cube volume = side × side × side, so side × side × side = 8.
- (a) 2 × 2 × 2 = 8, so the edge is 2 cm.
- (b) Original block volume = 88 + 8 = 96 cm³.
- (b) Base area = 6 × 4 = 24 cm², so h = 96 ÷ 24 = 4 cm.
- (c) Cubes with exactly 2 painted faces lie on the edges but not at the corners.
- (c) Each edge has 4 − 2 = 2 such cubes, and there are 12 edges, so 2 × 12 = 24 cubes.
The mistake this question tests
Calculation slip
Pupils often count the corner cubes, which have 3 painted faces, as if they had 2, or they forget to multiply the 2 cubes per edge by the 12 edges.
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