2021 THREE(e)

A lamp is suspended above the centre of a round table of radius $r$.

The height, $h$, of the lamp above the table is adjustable.

Point P is on the edge of the table.

At point P the illumination $I$ is directly proportional to the cosine of angle $\theta$ in the above diagram,
and inversely proportional to the square of the distance, $S$, to the lamp.

i.e. $I=\frac{k\cos \theta }{S^2}$, where $k$ is a constant.

Prove that the edge of the table will have maximum illumination when $h=\frac{r}{\sqrt{2}}$.

*You do not need to prove that your solution gives the maximum value.*

*You must use calculus and show any derivatives that you need to find when solving this problem.*