What does the $R$ in the formula $P=V^2/R$ for power represent? We know the following formula:
$$P=V^2/R.$$
I want to ask what the $R$ in the above formula represents. Is it the normal resistance of the resistor (load) or is it the increased resistance of the resistor (load) due to heat generation and consequently, increase in temperature of the resistor (load).
I am asking this because today we solved a question in class which read:

The resistance of a 240V and 100W electric lamp heats up from room temperature to operating temperature. As it heats up its resistance is increased by a factor of 16. What is the resistance of the lamp at room temperature?

The answer to the about problem was $36\:\Omega$ which obviously assumed that the $R$ in the above formula represents the resistance of the bulb when its temperature increases when its operating.
I think the answer should simply be $576 \:\Omega$. I think the $R$ in the formula simply shows what the resistance of the bulb will be when there is no increase in temperature, since it is derived using ohm's law (which works under constant physical conditions).
Please note that this is not a homework question and the teacher is probably not going to discuss it in class again (unless the real answer is not $36\:\Omega$ and I am correct).
 A: I think you should interpret the $100W$ power of the lamp as the power of the lamp at operating temperature. Hence, at operating temperature the resistance is $576\Omega$, and thus at room temperature the resistance will be a factor $16$ smaller.
A: The relation $P=V^2/R$ gives the power $P$ dissipated by a given element that's subjected to an external voltage $V$ and which presents an electrical resistance $R$ at the specified loading conditions.
Thus, if the resistance depends on the temperature, and the temperature depends on the dissipated power, then increasing the voltage will cause a change in the resistance that must be used.
As a general nomenclature framework, an ohmic conductor is a conductor which obeys Ohm's law to the letter: the current $I$ induced by a voltage $V$ is strictly porportional to the voltage (and therefore the resistance $R=V/I$ is independent of the applied voltage). As a general rule, whether a conductor is ohmic or not will depend on the material, the load (i.e. some materials will be ohmic at some applied voltages but not others), the conditions (so there might be e.g. temperature dependence) and the precision sought (so a material might be roughly ohmic but not if you care about high-precision measurements, and indeed if the precision you require is high enough then many materials will start to deviate from the ideal).
There are plenty of non-ohmic conductors around. Many useful examples come from semiconductor devices such as diodes, which might only conduct electricity in a single direction, or which might have threshold voltages below which they do not conduct at all. But the simplest non-ohmic conductors are lightbulb filaments, which heat up as the load increases, thereby increasing the resistance.

As regards your set-piece statement, though,

The resistance of a 240V and 100W electric lamp heats up from room temperature to operating temperature. As it heats up its resistance is increased by a factor of 16. What is the resistance of the lamp at room temperature?

it is very clear that (i) the expectation is that you are aware that resistance changes with the load, and (ii) the stated power rating of 100W under the applied voltage of 240V should be understood as representing the steady-state conditions in which the filament has warmed up (thus increasing its resistance by the stated factor). As such, those ratings are connected via $P=V^2/R$ at the increased resistance, not the nominal room-temperature value, which is correspondingly smaller.
A: The rule is
$$P=IV$$
Here the voltage, $V$, is the amount of electrical potential energy per unit charge, while the current $I$ is the rate at which charges pass through this energy difference. That’s why the product $IV$ represents the electrical energy exchanged per unit time in a circuit element.
The relation in your question is for the special case of an “ohmic” device where the current and voltage are proportional to each other, $I=V/R$.  If you have a device where the relationship between current and voltage is nonlinear, such a lamp whose filament changes resistance with temperature, or a transistor amplifier, or a plasma-discharge conductor, then it may or may not make sense to define an effective resistance at the instant you would like to compute the power.
A: $R$ in the equation is the resistance of the resistor when the power $P$ dissipation in the resistor is constant, i.e., at constant operating temperature.
Since the cold resistance is 36 Ohms, when the lamp is first switched on, there will be, assuming constant applied voltage, a theoretical inrush current of 240/36 = 6.85 A, or about 16 times the steady operating current of 0.42 A. This is a typical inrush factor for incandescent lamps.
Hope this helps.
A: The power rating of the bulb of 100W applies at its operating temperature. Hence when the bulb is at its operating temperature its effective resistance is 576 $\Omega $; this follows from the formula you quote.
However when the bulb cools to room temperature its resistance falls by a factor of 16 to 36 $\Omega$.
