# Hot air ballooning

**Classical Mechanics**Level 5

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 \(120^\circ\mbox{C}\), 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 \(300~\mbox{kg}\).
- The ambient pressure is \(1~\mbox{atm}=101,325~\mbox{Pa}\) and the temperature of the surrounding air is \(20^\circ\mbox{C}\).
- Air has a molar mass of \(\mu=29~\mbox{g/mol}\).

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