Radius of hot-air balloon

Question

The air in the hot air balloon is heated to 70 ° C. At the start, the surrounding atmospheric pressure is 98 kPa and the temperature is 28 ° C, the balloon is open from below, assuming a spherical shape and Mr air is 29.

What must be its radius to withstand a total load of 300 kg (not including the air)?

The radius is supposed to be 12,2 m but I always get 8,016 m. I'm desperate as I'm trying to solve this for days, I'm using the Archimedes law and pV=nRT. Thank you in advance for help.

Answer 1

Yes, Archimedes. Displace a volume of air equivalent to 300 kg.

First approximation, empty bag of volume (pi/6) D^3

air density about 1.3 kg/m^3

displacing (1.3 kg/m^3) (pi/6)D^3 = 300 kg

D^3= 300/(1.3)(pi/6) = 441 m^3

D = 8 m approximately.

YOU CAN DO THIS MORE ACCURATELY, BUT YOU SEE WE ARE IN THE BALLPARK ALREADY.