I have been reading about illuminance, illuminous flux, lumens, lux, irradiance, etc on wikipedia. I made this toy problem to test my knowledge. I am posting it here to test if I'm correct. (note, this really isn't a homework problem, I generated the problem myself).
Problem. Suppose we have a room of 5 by 5 by 5 metres, and we have a point light (i.e. a lamp) in the middle of the room. How many lumens must the lamp radiate, and how many Watts lightbulb (of a typical LED lamp) do we need, to produce the equivalent ambiant illuminance of daylight on a typical day?
Derivation. The SI unit of illumance is lux, and the ambient illuminance on a typical day is $10\;000$ lux (wikipedia). Assume the lightbulb radiates in all directions equally, and therefore produces $4\pi\cdot x/r^2\approx 12.5 x/r^2$ lux on the surface of a sphere of radius $r$ around it, where $x$ is the lumens of the lamp. If we approximate the room by a sphere of radius 2.5, then we have approximately a lux of $2x$ per square meter. So we need a luminous flux of approximately $5000$ lumens. Given that a typical LED lamp produces about 100 lumens per watt, this means we need about 50 watt LED lightbulb, or about 8 typical 6 watt lightbulbs. Answer: 5000 lumens, 50 watt.
Is this derivation correct? Does the answer pass the sanity check?