48. An electric lamp is suspended on a pulley above the center of a round table. At what height should this lamp be fixed to obtain the greatest illumination at the edges of the table?
48. Illuminance is given by the formula $E=\frac{k \sin \alpha}{A C^{2}}$, where $k$ is the proportionality coefficient; $A C$ is the distance from the lamp to the edge of the table. But $A C=\frac{R}{\cos \alpha}$, where $\alpha$ is the angle of inclination of $A C$ to the plane of the table. Then

$$
E=\frac{k \sin \alpha \cos ^{2} \alpha}{R^{2}} ;\left(\frac{E R^{2}}{2 k}\right)^{2}=\left(1-\cos ^{2} \alpha\right) \frac{\cos ^{2} \alpha}{2} \cdot \frac{\cos ^{2} \alpha}{2}
$$

Since the sum of the three factors is constant and equal to 1, the best illumination at point $C$ will be when $1-\cos \alpha=\frac{\cos ^{2} \alpha}{2}$, from which $\cos \alpha=\sqrt{\frac{2}{3}}, h=R \operatorname{tg} \alpha \approx 0.7 R$.

![](https://cdn.mathpix.com/cropped/2024_05_21_ec9ddf54a7c86775293eg-120.jpg?height=423&width=445&top_left_y=1856&top_left_x=383)

Fig. 30