$2\mathrm{.}42.$ Currently, recreational balloons are not filled with hydrogen or helium but with hot air; the pilot has a small propane burner to heat the air in the balloon. If the balloon is a sphere $20m$ in diameter, and if the total weight of balloon, pilot, passenger compartment, propane burner, propane tank, ropes, etc., is $200kg,$ what average Page $78$ temperature must the air in the balloon have to just barely lift the balloon? Assume that the air inside and outside the balloon both have atmospheric pressure and have equal molecular weights, $29g\frac{m}{m}ol.($The latter is slightly inaccurate because of the products of combustion inside the balloon; this inaccuracy is small.$)$