To answer these questions, you need to know the resistivity of tungsten as a function of temperature (or at least at the temperatures 1500ºC and 20ºC).
Using the citation below, I see that at 20ºC
ρ = 5.60e-5 Ω·m
and using the second citation, at 1500ºC (= 1773 K) the resistivity is about 8.75 times that value.
I assume that you have reference tables and/or formulae at your disposal that would give you the value of ρ at both temperatures. I would be surprised if both of my values are the ones you are expected to use. Check your references.
Anyways, the solution method from this point is:
You have the power and voltage:
resistance R = V²/P = (120V)² / 40W = 360 Ω
The area of the filament (assuming it is circular) is
A = πd²/4 = π(39e-6m)² / 4 = 1.2e-9 m²
R = ρL/A
and so at 1500ºC
360 Ω = 8.75*5.60e-8Ω·m * L / 0.0012m²
which solves to
L = 0.88 m
which seems awfully long, but that's what I get.
R = 5.6e-8Ω·m * 0.88m / 1.2e-9m² = 41 Ω
which you can also get by doing
R = 360Ω / 8.75 = 41 Ω
both of which assume that the 8.75 factor I'm using is correct.
That's the method; if the numbers I'm using are right, those are the answers.
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