Question

Marcus is organizing equipment in a large equipment shed. The shed measures **18 feet long, 10 1/2 feet high, and 12 feet wide**.



Marcus places 2 large storage crates along the back wall of the shed.



* **Crate A** measures **5 feet long, 6 feet high, and 4 1/2 feet wide**.

* **Crate B** measures **7 1/2 feet long, 5 feet high, and 3 feet wide**.



Marcus wants to fill the remaining space in the shed with identical smaller bins that measure **3 feet long, 2 1/2 feet high, and 3 feet wide**.



What is the greatest number of smaller bins Marcus can fit inside the shed?

Answer 1

First find how many small bins fit in the empty shed. Along the length 18 ÷ 3 = 6, along the width 12 ÷ 3 = 4, and in height 10.5 ÷ 2.5 = 4 layers, so 6·4·4 = 96 bins if there were no crates.

Now count how many of those 96 bin positions are blocked by the two crates (treating bins and crates as axis-aligned boxes):

- Crate A (5 × 6 × 4.5) overlaps 2 length-segments (0–3 and 3–6), 2 width-segments (0–3 and 3–6), and 3 height-layers (0–2.5, 2.5–5, 5–7.5), so it blocks 2·2·3 = 12 bin positions.
- Crate B (7.5 × 5 × 3) overlaps 4 length-segments (3–6, 6–9, 9–12, 12–15), 1 width-segment (0–3), and 2 height-layers (0–2.5, 2.5–5), so it blocks 4·1·2 = 8 bin positions.
- One set of bin positions (the cells in length 3–6, width 0–3, height layers 0 and 1) was counted in both crates, so subtract that double-counted 2.

Total blocked = 12 + 8 − 2 = 18. So the maximum number of small bins that can fit is 96 − 18 = 78.