= Solution
The target liquid-<water> <mass> is
$$
m_* =\frac{4\pi}{3}(10^3)(15\times10^{-6})^3
=1.414\times10^{-11}\,\mathrm{kg}.
$$
The <thin-plate ice-growth time> uses the prescribed ice <mass> law, which therefore requires $r_{i*}=(m_*/\alpha)^{1/3}=45.31\,\mu\mathrm m$, not the same <radius> as the <water> droplet. With $C=2r_i/\pi$ and $F_i=F_k^i+F_d^i$,
$$
3\alpha r_i^2\frac{dr_i}{dt}=\frac{8r_i(S^i-1)}{F_i},\qquad
\frac{d(r_i^2)}{dt}=\frac{16(S^i-1)}{3\alpha F_i}.
$$
Hence, for negligible seed <mass> and constant conditions,
$$
\boxed{t_i=\frac{3\alpha F_i r_{i*}^2}{16(S^i-1)}
=\frac{3(152)(2.1\times10^7)(45.31\times10^{-6})^2}{16(0.05)}
\simeq24.6\,\mathrm s.}
$$
This is about $8.24$ times faster than the liquid estimate to the same <mass>. Both laws have $dm/dt\propto m^{1/3}$ under the given shape assumptions. Ice benefits here from five times the <atmospheric supersaturation> and a larger geometrical <diffusion-capacitance analogy>[capacitance] for the same <mass>, partly offset by its slightly greater transport resistance. The <mass> coefficient $\alpha$ encodes the assumed plate shape; replacing it by bulk ice <density> would change the supplied model.
In a mixed-phase <cloud>, saturation vapour <pressure> over ice is lower than over supercooled liquid <water>. Air can therefore be subsaturated with respect to liquid yet supersaturated with respect to ice: droplets evaporate while ice crystals grow, transferring <water> through the vapour phase. This is the <Wegener-Bergeron-Findeisen process>[Wegener–Bergeron–Findeisen mechanism]. Subsequent aggregation, <riming> and melting can produce <atmospheric precipitation> much faster than <condensation> alone.
The large difference is not a universal fixed ice/liquid time ratio. <Atmospheric supersaturation> evolves, latent heating increases thermal resistance, crystal habits and <diffusion-capacitance analogy>[capacitance] change, and surface attachment or ventilation may matter. Ice nucleation and the availability of freezing nuclei also precede the calculated growth. The boxed time describes the specified already-present crystals under maintained conditions, not a complete prediction of the time until rain begins.
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