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 could fit with no crates.

- Along the length: 18 ÷ 3 = 6 bins
- Along the width: 12 ÷ 3 = 4 bins
- In height: 10.5 ÷ 2.5 = 4 layers

So without crates: 6 × 4 × 4 = 96 bins.

Now count the grid cells (3 × 3 × 2.5 blocks) blocked by the crates. (Crates are flush against the back wall, and rotating them doesn’t change the number of 3-ft/3-ft/2.5-ft grid cells they overlap.)

- Crate A is 5 × 4.5 × 6, so it spans ceil(5/3)=2 cells in length, ceil(4.5/3)=2 cells in width, and ceil(6/2.5)=3 layers in height → blocks 2×2×3 = 12 bin spaces.
- Crate B is 7.5 × 3 × 5, so it spans ceil(7.5/3)=3 cells in length, ceil(3/3)=1 cell in width, and ceil(5/2.5)=2 layers → blocks 3×1×2 = 6 bin spaces.

Total blocked = 12 + 6 = 18 spaces.

Thus the greatest number of small bins that can fit = 96 − 18 = 78.