Answer
When: $T_{atm}=12^{\circ}C$
$T_{ballon}=308.26K$
When: $T_{atm}=25^{\circ}C$
$T_{ballon}=323.47K$
Work Step by Step
When: $T_{atm}=12^{\circ}C$
$V_{ballon}=\frac{4\pi r^3}{3}=\frac{4\pi (9m)^3}{3}=3053.63m^3$
$\rho_{coolair}=\frac{P}{RT}=\frac{93kPa}{0.287\frac{kPam^3}{kgK}*(12+273.15)K}=1.1364\frac{kg}{m^3}$
$F_{B}=\rho_{coolair}gV_{ballon}=1.1364\frac{kg}{m^3}*9.81\frac{m}{s^2}*3053.63m^3=34.04kN$
$F_{B}=W_{people}+W_{cages}+W_{hotair}$
$W_{hotair}=34.04kN-9.81\frac{m}{s^2}*(2*70kg+120kg)=31.49kN$
$m_{hotair}=\frac{W_{hotair}}{g}=\frac{31.49kN}{9.81\frac{m}{s^2}}=3209.93kg$
$T=\frac{PV}{mR}=\frac{93kPa*3053.63m^3}{3209.93kg*0.287\frac{kPam^3}{kgK}}=308.26K$
When: $T_{atm}=25^{\circ}C$
$\rho_{coolair}=\frac{P}{RT}=\frac{90kPa}{0.287\frac{kPam^3}{kgK}*(25+273.15)K}=1.0868\frac{kg}{m^3}$
$F_{B}=\rho_{coolair}gV_{ballon}=1.0868\frac{kg}{m^3}*9.81\frac{m}{s^2}*3053.63m^3=32.56kN$
$W_{hotair}=32.56kN-9.81\frac{m}{s^2}*(2*70kg+120kg)=30.01kN$
$m_{hotair}=\frac{W_{hotair}}{g}=\frac{30.01kN}{9.81\frac{m}{s^2}}=3059.06kg$
$T=\frac{PV}{mR}=\frac{93kPa*3053.63m^3}{3059.06kg*0.287\frac{kPam^3}{kgK}}=323.47K$
