Hot air ballooning

You are a maker of hot air balloons and wish to construct a balloon that will lift you and a friend to places unknown. Your standard balloon design is a perfectly spherical, nylon fabric balloon with a small hole on the bottom for the burner that heats the air. Nylon starts to degrade if the air temperature exceeds 120C120^\circ\mbox{C}, so you don't want to exceed this temperature for your hot air. What is the minimum radius in meters you need for your balloon?

Details and assumptions

  • You, your friend, the balloon, the burner, and the basket have a total mass of 300 kg300~\mbox{kg}.
  • The ambient pressure is 1 atm=101,325 Pa1~\mbox{atm}=101,325~\mbox{Pa} and the temperature of the surrounding air is 20C20^\circ\mbox{C}.
  • Air has a molar mass of μ=29 g/mol\mu=29~\mbox{g/mol}.

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