Siti had a rectangular block of foam for a school project. The block is 6 cm long and 5 cm wide. She cut a cube out of one top corner of the block. The edges of the cube lie along the edges of the block. Then she painted all the faces of the remaining block yellow. The volume of the cube that was cut out is 64 cm³.
- Find the length of one edge of the cube that was cut out.
- The volume of the remaining block is 116 cm³. Find the height of the original rectangular block.
- Siti then cut the remaining block into 1-cm cubes. How many of these small cubes have exactly 3 faces painted yellow?
Need a hint?
For (c), look at the block layer by layer. In each layer, count how many faces of each small cube are on the outside.
Worked answer
- (a) Find the number that gives 64 when multiplied by itself 3 times: 4 × 4 × 4 = 64, so the edge is 4 cm.
- (b) Add the cube back to find the original volume: 116 + 64 = 180 cm³.
- (b) The base area is 6 × 5 = 30 cm², so the height is 180 ÷ 30 = 6 cm.
- (c) The block is 6 cm high and the cube is 4 cm tall, so the bottom 2 layers are full 6 by 5 layers and the top 4 layers are L-shaped.
- (c) Bottom layer: the 4 corner cubes each have 2 side faces and the bottom face painted, giving 4 cubes.
- (c) Second layer: only the cube right under the cut-out corner has its top and 2 side faces painted, giving 1 cube.
- (c) Third, fourth and fifth layers: only the cube at the end of the narrow (1-cube-wide) arm of the L has 3 faces painted: 2 outside faces and 1 face on the cut. That gives 3 layers × 1 = 3 cubes.
- (c) Top layer: 6 cubes have 2 side faces plus the top face painted.
- (c) Total = 4 + 1 + 3 + 6 = 14 cubes.
The mistake this question tests
Calculation slip
Many pupils count only the corners of the original block (8 − 1 = 7). They forget that cutting out the cube makes new cubes with 3 painted faces along the edges of the cut.
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