3.46. Find the angle between the slant height and the height of a cone, the lateral surface area of which is the mean proportional between the area of the base and the total surface area.
3.46. According to the condition, $S_{\text {osn }}: S_{\text {bok.pov }}=$ $=S_{\text {bokpov }}: S_{\text {poln }} ;$ we need to find $\angle C B O=$ $=x$ (Fig. 3.48). Let $O C=R$; then $S_{\text {osn }}=$ $=\pi R^{2}, S_{\text {bok.pov }}=\pi R l$, where $l=B C$. Therefore,

$$
\frac{\pi R^{2}}{\pi R l}=\frac{\pi R l}{\pi R^{2}+\pi R l}, \text { or } \frac{R}{l}=\frac{l}{R+l}
$$

Since $R=l \sin x$, then

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-528.jpg?height=534&width=419&top_left_y=106&top_left_x=735)

Fig. 3.48

$$
\frac{\sin x}{1}=\frac{1}{\sin x+1} ; \sin ^{2} x+\sin x-1=0 ; \sin x=\frac{-1 \pm \sqrt{5}}{2}
$$

Only the value $\sin x=\frac{\sqrt{5}-1}{2}$ is valid, from which $x=\arcsin \frac{\sqrt{5}-1}{2}$.