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 count how many small bins fit in the empty shed if nothing else were inside. Along the shed's length 18 ÷ 3 = 6 bins, along the width 12 ÷ 3 = 4 bins, and in height 10.5 ÷ 2.5 = 4 layers. So total possible bin positions = 6·4·4 = 96.

Now count how many of those positions are blocked by the two crates. Treat the bin positions as a 3×3×2.5 grid (cells 3×3×2.5) and count cells that a crate intersects.

Crate A (5 × 6 × 4.5) reaches into 2 length-cells (0–5 intersects the 0–3 and 3–6 length cells), 2 width-cells (0–4.5 intersects 0–3 and 3–6 width cells), and 3 height-cells (0–6 intersects the 0–2.5, 2.5–5, and 5–7.5 height cells). So it blocks 2·2·3 = 12 bin positions.

Crate B (7.5 × 5 × 3) reaches into 3 length-cells (0–7.5 → 0–3, 3–6, 6–9), 2 width-cells (4.5–7.5 → 3–6 and 6–9), and 2 height-cells (0–5 → 0–2.5 and 2.5–5). So it blocks 3·2·2 = 12 bin positions.

The two crates do not overlap in width (they meet only at a boundary), so blocked positions add: 12 + 12 = 24.

Therefore the greatest number of small bins that can fit = 96 − 24 = 72.