MHB What Height Should a Lamp Be Above a Round Table for Maximum Edge Lighting?

[leprofece]

At what height above the center of a round table of RADIUS r should be an electric lamp to make the edge lighting Maximun?. (Note: llumlnation is expressed for the forrmula I = k sen f/ d ² where f is the angle of inclination of rays, d is the distance from the light source to the illuminated surface and K as the intensity of the light source

Answer r/(sqrt 2)

 

[MarkFL]

You really should be able to at least show an attempt. I would begin by observing:

$$\sin(f)=\frac{h}{d}$$

Hence:

$$I=k\frac{h}{d^3}$$

Now, by Pythagoras, we know:

$$r^2+h^2=d^2$$

So, can you now state $I$ as a function of the variable $h$ and the constant $r$? Then, differentiate with respect to $h$ and equate the result to zero, then solve for $h$, and the result you cite will follow.