Answer
When: $T_{atm}=15^{\circ}C$
$T_{ballon}=306.65K$
When: $T_{atm}=30^{\circ}C$
$T_{ballon}=323.67K$
Work Step by Step
When: $T_{atm}=15^{\circ}C$
$V_{ballon}=\frac{4\pi r^3}{3}=\frac{4\pi (10m)^3}{3}=4188.79m^3$
$\rho_{coolair}=\frac{P}{RT}=\frac{90kPa}{0.287\frac{kPam^3}{kgK}*(15+273.15)K}=1.0883\frac{kg}{m^3}$
$F_{B}=\rho_{coolair}gV_{ballon}=1.0883\frac{kg}{m^3}*9.81\frac{m}{s^2}*4188.79m^3=44.72kN$
$F_{B}=W_{people}+W_{cages}+W_{hotair}$
$W_{hotair}=44.72kN-9.81\frac{m}{s^2}*(3*65kg+80kg)=42.02kN$
$m_{hotair}=\frac{W_{hotair}}{g}=\frac{42.02kN}{9.81\frac{m}{s^2}}=4283.61kg$
$T=\frac{PV}{mR}=\frac{90kPa*4188.79m^3}{4283.61kg*0.287\frac{kPam^3}{kgK}}=306.65K$
When: $T_{atm}=30^{\circ}C$
$\rho_{coolair}=\frac{P}{RT}=\frac{90kPa}{0.287\frac{kPam^3}{kgK}*(30+273.15)K}=1.0344\frac{kg}{m^3}$
$F_{B}=\rho_{coolair}gV_{ballon}=1.0344\frac{kg}{m^3}*9.81\frac{m}{s^2}*4188.79m^3=42.51kN$
$W_{hotair}=42.51kN-9.81\frac{m}{s^2}*(3*65kg+80kg)=39.81kN$
$m_{hotair}=\frac{W_{hotair}}{g}=\frac{39.81kN}{9.81\frac{m}{s^2}}=4058.33kg$
$T=\frac{PV}{mR}=\frac{90kPa*4188.79m^3}{4058.33kg*0.287\frac{kPam^3}{kgK}}=323.67K$
