Crate Capacity: When the Volume Shortcut Overcounts
The packaging team at a school supply company is choosing a shipping box for 2-inch cube sample kits. Box A measures 8 inches by 6 inches by 5 inches. Box B measures 7 inches by 6 inches by 5 inches. A volunteer computes volume and reports that Box A holds 240 ÷ 8 = 30 cubes and that Box B holds 210 ÷ 8 = 26.25, "so about 26 cubes." Box B costs less per carton, and the team wants each carton to ship 20 kits. Before ordering 500 Box B cartons, the team leader asks for a check: can the cubes really be arranged to fill the space the volume calculation assumes? She is especially suspicious of Box B's 7-inch side. The team also wants to know how much empty space is left inside each box after the cubes are packed in complete layers.
1. How many 2-inch cubes fit into each box when the cubes are packed in complete layers? Show the packing reasoning and explain why the volume division gives a different answer for Box B.
Read the worked answer
Pack whole cubes along each edge. A leftover strip shorter than 2 inches cannot hold a cube.
Box A: 8 ÷ 2 = 4 cubes along the length; 6 ÷ 2 = 3 along the width; 5 ÷ 2 = 2 layers with 1 inch left over. Total: 4 × 3 × 2 = 24 cubes.
Box B: 7 ÷ 2 = 3 cubes with 1 inch left over; 6 ÷ 2 = 3; 5 ÷ 2 = 2. Total: 3 × 3 × 2 = 18 cubes.
The volume method (total volume ÷ 8) treats leftover space as if it could be melted together into extra cubes. In Box A, 240 ÷ 8 = 30 assumes the 1-inch strip across the top becomes 6 more cubes; it cannot. Box B is worse: 210 ÷ 8 = 26.25 ignores a 1-inch strip on the 7-inch side. Volume division is exact only when every edge is a multiple of the cube's edge length.
2. Which box should the team order if each carton must hold 20 kits? Support the choice with the empty space left in each packed box.
Read the worked answer
Capacity test: Box A holds 24 cubes, and 24 ≥ 20, so it works. Box B holds 18, and 18 < 20 - two kits would have no place. Box B fails.
Empty space:
- Box A: volume 8 × 6 × 5 = 240 in³. Cubes take 24 × 8 = 192 in³. Empty: 240 − 192 = 48 in³.
- Box B: volume 7 × 6 × 5 = 210 in³. Cubes take 18 × 8 = 144 in³. Empty: 210 − 144 = 66 in³.
Box A is the answer on both counts: it reaches the 20-kit target and wastes less space per carton. The lower Box B price is misleading, because shipping two loose kits per carton would erase the savings.
3. State the general condition under which dividing a box's volume by a cube's volume gives the exact number of cubes, and use it to identify which box in this problem would have been safe to judge by volume alone.
Read the worked answer
The division is exact only when each edge of the box is a whole-number multiple of the cube's edge length. Then no partial strips exist and every cubic inch is usable.
Check Box A: 8, 6, and 5 against 2 - 8 and 6 are multiples of 2, but 5 is not (5 ÷ 2 = 2.5). So Box A is not safe either; it holds 24, not 30.
Check Box B: 7 is not a multiple of 2 and 5 is not, so it is definitely unsafe.
Neither box qualifies here, which is the real lesson: a faster-looking computation giving a neat decimal is not evidence that the answer is right. The reliable test is to ask whether a whole number of cubes fits along each edge before trusting a volume division.