Question

A large He balloon of volume 1425 m³ is used to lift a payload of 400 kg. Assume that the balloon maintains constant radius as it rises. How high does it rise ?

Answer 1

your question is incomplete. A complete question is ----> A large He balloon of volume 1425 m³ is used to lift a payload of 400 kg. Assume that the balloon maintains constant radius as it rises. How high does it rise ?($y_0$=8000m,$\rho_{He}=0.18kg/m^3$)

Volume of He balloon, V = 1425 m³
mass is used to lift a payload , m = 400kg
total mass of balloon , M = $m+V\rho_{He}$
M = 400 kg + 1425 × 0.18
M = 400kg + 256.5 kg
M = 656.5 kg

so, density , $\rho=\frac{M}{V}$
= 656.5/1425
= 0.46 kg/m³

we know, rate of change of decreases of density per unit height from the sea level is directly proportional to density
e.g., $-\frac{d\rho}{dy}\propto\rho$

$\frac{d\rho}{dy}=-k\rho$

$\int\limits^{\rho}_{\rho_0}{\frac{1}{\rho}}\,d\rho=-k\int\limits^y_0\,dy$

$log_e(\rho-\rho_0)=-ky$

put k = 1/y0

$\rho=\rho_0e^{-\frac{y}{y_0}}$

now, put the values of all given data.
so, y = 8000 m = 8km