In this problem, we are required to apply the concept of buoyant force.

Buoyant force in hot air balloons:

$\overline{){{\mathbf{F}}}_{{\mathbf{B}}}{\mathbf{=}}{\mathbf{(}}{{\mathbf{\rho}}}_{{\mathbf{a}}}{\mathbf{-}}{{\mathbf{\rho}}}_{{\mathbf{T}}}{\mathbf{)}}{{\mathbf{V}}}_{{\mathbf{balloon}}}{\mathbf{g}}}$ ρ_{a} is the density of air and ρ_{T} is the density of hot air.

Hot air balloons float by heating the air inside an inflatable compartment. Consider such a balloon which has an inflatable compartment of maximum volume 2050 m^{3} and a basket with passengers of total mass 390 kg.

Part (a) How hot, in degrees Celsius, would the air inside the balloon have to get in order for the balloon to lift off the ground? Assume the molar mass of air is 28.97 g/mol and its density is 1.20 kg/m^{3}

Part (b) When the balloon reaches the highest point of its flight (around 2000 m), the air pressure has dropped to 80% of the pressure at ground level, and the density has dropped to 85%. What temperature, in degrees Celsius, must the air in the balloon be to keep it floating at this altitude?

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