Calculating the Radius of a Hot Air Balloon to Withstand a Load of 300 kg

[Lenoshka]

Homework Statement

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)? [12,2 m]

Relevant Equations

Archimedes law, buoyant force

The result is supposed to be 12,2 m but every time I get 8,016 m... I used for example this formula >r=m/[(density of air-density of hot air)*(4/3)*pi]
For density I used > rho=(p*M)/(R*T)

Am I forgetting something? Thanks in advance.

 

[Charles Link]

The buoyant force comes from the weight/mass of the air at ambient temperature that is displaced, but don't forget to add the weight of the heated air to the load of 300 kg.

 

[Orodruin]

Homework Statement:: 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)? [12,2 m]
Relevant Equations:: Archimedes law, buoyant force

r=m/[(density of air-density of hot air)*(4/3)*pi]

You forgot a power of 1/3 here, but otherwise it looks correct to me.

The buoyant force comes from the weight/mass of the air at ambient temperature that is displaced, but don't forget to add the weight of the heated air to the load of 300 kg.

That is where the density of the hot air comes from in the (corrected) formula above:
$$
r = \left(\frac{3m}{4\pi (\rho_{\rm air} - \rho_{\rm hotair})}\right)^{1/3}
$$
If he just took the buoyant force only the density of the cold air would be in the formula.

Plugging the numbers into Octave, I get 8.0 m (the input data really does not support using more significant digits).