Thermodynamics – Hot-air balloon

1. The problem statement, all variables and given/known data
The mass of a hot-air balloon and its cargo (not including the air inside) is 200kg. The air outside is at 10C and 101kPa. The volume of the balloon is 400m^3. To what temperature must the air in the balloon be warmed before the balloon will lift off? (The density of air a 10C is 1.25kg/m^3).

2. Relevant equations
and maybe some other basic thermodynamics.

3. The attempt at a solution

[itex]\sum F = _{}\rho_{inside}Vg – _{}\rho_{outside}Vg – mg = 0[/itex]
Simplifies to
[itex]m= V(_{}\rho_{inside} – {}\rho_{outside})[/itex]

Assuming the volume is the same…. But I can’t really wrap my head around it. Would the volume be the same? I have no idea what to do next and I’m not even sure that I’m doing it right.

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