= Solution
Let $r_s^2=N^{-1}\int r^2n(r)\,dr$. The <Cloud droplet effective radius>[effective droplet radius] and <Adiabatic cloud droplet moments>[shape factor] give $r_e=r_v^3/r_s^2$ and $k^{1/3}=r_v/r_e$, so $r_s^2=k^{1/3}r_v^2$. If the <Particle extinction coefficient>[extinction] efficiency is approximately independent of <Cloud droplet>[droplet] size over the distribution,
$$
\sigma_{\rm ext}=\pi Q_{\rm ext}Nr_s^2
=\pi Q_{\rm ext}k^{1/3}\left(\frac{3c_w}{4\pi\rho_w}\right)^{2/3}N^{1/3}h^{2/3}.
$$
For visible <Wavelength>[wavelengths] and <Cloud droplet>[droplets] large compared with the <wavelength>, <geometrical optics> makes $Q_{\rm ext}\simeq2$ a useful approximation. Retaining it as the specified constant, and taking $N,k,c_w$ constant over the <cloud>,
$$
\boxed{\tau_c=\int_0^H\sigma_{\rm ext}\,dh
=\frac35\pi Q_{\rm ext}k^{1/3}
\left(\frac{3c_w}{4\pi\rho_w}\right)^{2/3}N^{1/3}H^{5/3}.}
$$
The $3/5$ comes from integrating $h^{2/3}$; replacing the whole <cloud> by its cloud-top <Particle extinction coefficient>[extinction] would incorrectly omit it. The effective-radius relation treats the distribution shape through $k$, without assuming all <Cloud droplet>[droplets] have the same size. Liquid phase, negligible vertical variation in that <Adiabatic cloud droplet moments>[shape factor], and negligible departures from the adiabatic water profile are essential assumptions.
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