3.96. A cube is inscribed in a cone (one of the cube's faces lies in the base plane of the cone). The ratio of the height of the cone to the edge of the cube is $k$. Find the angle between the slant height and the height of the cone.
3.96. By the condition, $SAB$ is a cone, $SO$ is its height, $\operatorname{CDEFC}_{1}D_{1}E_{1}F_{1}$ is a cube inscribed in the cone, $C_{1}, D_{1}, E_{1}, F_{1}$ are points lying on the lateral surface of the cone, $SO: CC_{1}=k$ (Fig. 3.96). In $\Delta \mathrm{SO}_{1}C_{1}$, we have $\operatorname{ctg} \angle ASO = \frac{SO_{1}}{O_{1}C_{1}} = \frac{2(SO - x)}{x \sqrt{2}}$, where $x = CC_{1}$. From this, we find

$$
\operatorname{ctg} \angle ASO = \sqrt{2}\left(\frac{SO}{x} - 1\right) = \sqrt{2}(k - 1)
$$

Thus, $\angle ASO = \operatorname{arcctg}(\sqrt{2}(k - 1))$.

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-562.jpg?height=523&width=342&top_left_y=1246&top_left_x=152)

Fig. 3.95

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-562.jpg?height=488&width=360&top_left_y=1279&top_left_x=718)

Fig. 3.96

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-563.jpg?height=534&width=510&top_left_y=104&top_left_x=45)

Fig. 3.97