## Task 4 - 090514

In the workshop, students make building blocks that have the shape of cuboids, and for each building block, three edges running in different directions are $55 \mathrm{~mm}, 55 \mathrm{~mm}$, and $70 \mathrm{~mm}$ long.

For better storage, these building blocks are packed into cuboid building boxes (with a sliding lid), whose inner dimensions (measured in the three directions of the edges) are $0.33 \mathrm{~m}, 2.2 \mathrm{~dm}$, and $21 \mathrm{~cm}$.

Calculate the maximum number of building blocks that can be stacked in six of these building boxes!
The volume of the box for storing the building blocks is

$$
V_{\text {Box }}=33 \mathrm{~cm} \cdot 22 \mathrm{~cm} \cdot 21 \mathrm{~cm}=15246 \mathrm{~cm}^{3}
$$

The volume of one building block is:

$$
V_{\text {Building Block }}=5.5 \mathrm{~cm} \cdot 5.5 \mathrm{~cm} \cdot 7 \mathrm{~cm}=211.75 \mathrm{~cm}^{3}
$$

Theoretically, $15246: 211.75=72$ building blocks can fit into the storage box.

However, it needs to be checked whether the building blocks can be inserted into the box without any gaps.

With the edge lengths of $55 \mathrm{~mm}$ and $55 \mathrm{~mm}$, all gaps on the base area of the box can be filled by arranging $6 \cdot 5.5$ $\mathrm{cm}=33 \mathrm{~cm}$ in length and $4 \cdot 5.5 \mathrm{~cm}=22 \mathrm{~cm}$ in width. Thus, $6 \times 4=24$ building blocks lie on the base area.

Since the height of the box is $21 \mathrm{~cm}$ and one building block has a height of $7 \mathrm{~cm}$, 3 layers of building blocks can be inserted, meaning a total of 72 building blocks.