Why is it necessary that all three thermometers were at approximately the same distance from the light bulb

I'm assuming it was to keep the data consistent? The further you are from a heat source the less heat will get to you as the temperature tries to reach equilibrium and the waves start to spread out, so you should keep everything the same distance to get consistent results. I don't have any information so this is just my assumption

Pe = 1/2k(x)^2  and .05n/m = .05x10^-9m = x