Hot air balloons float in the air because of the difference in density between c

Question

Hot air balloons float in the air because of the difference in density between cold and hot air. Consider a balloon in which the mass of the pilot basket together with the mass of the balloon fabric and other equipment is mb. The volume of the hot air inside the balloon is V1 and the volume of the basket, fabric, and other equipment is V2. The absolute temperature of the cold air outside the balloon is Tc and its density is c . The absolute temperature of the hot air at the bottom of the balloon is Th (where Th>Tc). The balloon is open at the bottom, so that the pressure inside and outside the balloon is the same here. Assume that we can treat air as an ideal gas. Use g for the magnitude of the acceleration due to gravity

Question D is

Part D

For the balloon to float, what is the minimum temperature Tmin of the hot air inside it?

Express the minimum temperature in terms of Tc , V1, V2, mb, and c

If you could please show me the step by step process of how to get the answer that would be great. Thanks

Explanation / Answer

The balloon is open at the bottom, so that the pressure inside and outside the balloon remains the same.

The minimum temparature required is

T min = Tc*(V1/((V1+V2)-mb/c))

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