Lamp height above the street

1. The problem statement, all variables and given/known data
Above the center of the street a lamp is hanged. A lamp radiates equally in all directions. How high does the lamp have to be so the edge of the road is as bright as possible?

2. Relevant equations

3. The attempt at a solution

I’m kinda lost here…

##j=\frac{P}{A}## where ##A=4\pi r^2## therefore

##j=\frac{P}{4\pi r^2}##.

Now the idea is to get ##j(h)## but nothing seems to work… For example: Lets say that the street is ##2a## wide, therefore ##r^2=h^2+a^2##.

##j=\frac{P}{4\pi (h^2+a^2)}##

##\frac{\partial j}{\partial h}=-\frac{P}{4\pi}\frac{2h}{(h^2+a^2)^2}=0##…

Hmm, what am I missing here?

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