A student is reading a book located near the edge of a circular...

Let the height of the source above the center of the table be $h$.

The distance from the source to the book at the edge of the table is $r = \sqrt{R^2 + h^2}$.

The illuminance $E$ at the book's position is given by $E = \dfrac{I \cos \theta}{r^2}$, where $I$ is the luminous intensity of the source and $\theta$ is the angle of incidence.

From the geometry, $\cos \theta = \dfrac{h}{r} = \dfrac{h}{\sqrt{R^2 + h^2}}$.

Substituting the values of $r$ and $\cos \theta$, we get:

$E = \dfrac{I h}{(R^2 + h^2)^{3/2}}$

For maximum illuminance, $\dfrac{dE}{dh} = 0$.

$\dfrac{dE}{dh} = I \left[ \dfrac{1 \cdot (R^2 + h^2)^{3/2} - h \cdot \dfrac{3}{2} (R^2 + h^2)^{1/2} \cdot 2h}{(R^2 + h^2)^3} \right] = 0$

$(R^2 + h^2)^{3/2} - 3h^2 (R^2 + h^2)^{1/2} = 0$

Dividing by $(R^2 + h^2)^{1/2}$, we obtain:

$R^2 + h^2 - 3h^2 = 0$

$2h^2 = R^2$

$h = \dfrac{R}{\sqrt{2}}$