4B. A cube with an edge of 3 cm has all 8 of its corners cut off so that the newly formed body has 24 vertices and all its edges are of equal length. Determine the volume of the newly formed body.
Solution. From the equations

$$
b+2 c=a=3 \text { and } b=c \sqrt{2} \text {, }
$$

we get that $c=\frac{3(2-\sqrt{2})}{2} \mathrm{~cm}$. The volume of each of the 8 tetrahedra is

$$
V_{T}=\frac{c^{3}}{6}=\frac{9(10-7 \sqrt{2})}{8} \mathrm{~cm}^{3}
$$

The volume of the cube is

![](https://cdn.mathpix.com/cropped/2024_06_05_6152dee85d0865b6f018g-7.jpg?height=454&width=449&top_left_y=1782&top_left_x=1030)

$$
V_{K}=a^{3}=27 \mathrm{~cm}^{3}
$$

Finally, the volume of the body is

$$
V=V_{K}-8 V_{T}=63(\sqrt{2}-1) \mathrm{cm}^{3}
$$

## 4th Year

1AB. The lines $x=0$ and $4 x=3 y$ are tangents to a circle located in the first quadrant and whose center lies on the curve $y=2 x^{2}-27$. Find the radius of the circle.