= Solution
Using the <adiabatic cloud droplet moments>, define $N=\int n(r)\,dr$ and the <Adiabatic cloud droplet moments>[mean volume radius] by $r_v^3=N^{-1}\int r^3n(r)\,dr$. Summing the liquid mass of spherical <Cloud droplet>[droplets] gives
$$
w=\frac{4\pi\rho_w}{3}\int r^3n(r)\,dr
=\frac{4\pi\rho_w}{3}Nr_v^3.
$$
Therefore, with the shallow-cloud <liquid-water content> $w=c_wh$,
$$
\boxed{r_v(h)=\left(\frac{3c_wh}{4\pi\rho_wN}\right)^{1/3}.}
$$
The growth is $h^{1/3}$ at fixed <Number density>[number concentration], and the radius scales as $N^{-1/3}$ at fixed <Liquid-water content>[water content]. The dimensions are consistent: $c_w$ has units $\mathrm{kg\,m^{-4}}$, so $c_wh/(\rho_wN)$ has units $\mathrm{m^3}$. The formula is a third-moment radius; a monodisperse distribution is not required. Fixed $N$ per volume is the shallow-cloud approximation, not exact conservation of number per parcel volume during an arbitrarily deep expansion.
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