= Solution
Treat the target as an absorbing sphere of effective capture radius $r_0$, surrounded by a uniform dilute reservoir of <RNA polymerase> with <number density> $C_\infty$. The steady <diffusion equation>, <spherical symmetry>, and boundary conditions give
$$
\nabla^2 C=0,\qquad C(r_0)=0,\quad C(\infty)=C_\infty,
\qquad C(r)=C_\infty\left(1-\frac{r_0}{r}\right).
$$
By <Fick's laws>, the inward capture rate per target is $4\pi r_0^2D_A C'(r_0)=4\pi D_A r_0C_\infty$. Multiplying by the target <number density> and comparing with the bimolecular <rate law> therefore yields the <Smoluchowski association rate>
$$
\boxed{k_a=4\pi D_A r_0}.
$$
Its units here are $\mathrm{m^3\,s^{-1}}$, since the <concentrations> count particles rather than <moles>. The absorbing boundary assumes every encounter is productive; orientation constraints would reduce the effective <association rate constant>.
The <Stokes–Einstein relation>, for <RNA polymerase> radius $a=15\,\mathrm{nm}$ at an illustrative <temperature> $T=300\,\mathrm K$, gives
$$
D_A=\frac{k_BT}{6\pi\eta a}=1.46\times10^{-12}\,\mathrm{m^2\,s^{-1}}.
$$
Neither cell volume nor the effective $r_0$ is specified, so a unique numerical time is unavailable. A representative <Escherichia coli> volume is $V=1\,\mathrm{\mu m^3}=10^{-18}\,\mathrm{m^3}$. The $40$-<base pair> target has a <DNA> contour extent of about $40(0.34)=13.6\,\mathrm{nm}$; using this as an order-of-magnitude capture radius gives $k_a\simeq2.50\times10^{-19}\,\mathrm{m^3\,s^{-1}}$. For $M=3000$ independent searchers, the target's approximately constant encounter hazard is $Mk_a/V$, hence
$$
\boxed{\tau_3\simeq\frac{V}{Mk_a}=1.33\,\mathrm{ms}
\left(\frac{V}{1\,\mathrm{\mu m^3}}\right)
\left(\frac{13.6\,\mathrm{nm}}{r_0}\right)
\left(\frac{300\,\mathrm K}{T}\right)}.
$$
Using half the target's contour extent for $r_0$ instead gives about $2.7\,\mathrm{ms}$; the spherical capture model is not exact target geometry. One searcher's mean would be about $4\,\mathrm s$ under the first convention, whereas the first arrival among $3000$ is much faster.
If only a fraction $f_3=0.1$ of each searcher's time is available for three-dimensional <diffusion>, and nonspecific <DNA> binding neither scans nor delivers the target, the effective searching population is $f_3M$. Thus $\boxed{\tau\simeq\tau_3/f_3=10\tau_3}$, about $13\,\mathrm{ms}$ under the stated estimate. This assumes stationarity, well-mixed searchers and no additional target-access route; nonspecific binding alone is sequestration, not <facilitated diffusion>.
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