= Solution
The mean witness gradient must be $112.5\,\mathrm{GeV}/1.1\,\mathrm m\simeq102\,\mathrm{GV\,m^{-1}}$. To turn this into a <density> estimate, assume an accelerating wake amplitude of order the cold <plasma> wave-breaking scale,
$$
E_{\rm w}\sim\frac{m_ec\omega_p}{e},\qquad
\omega_p^2=\frac{n_0e^2}{\varepsilon_0m_e}.
$$
Eliminating $\omega_p$ gives
$$
\boxed{n_0\sim\frac{\varepsilon_0E_{\rm w}^2}{m_ec^2}
\simeq1.1\times10^{24}\,\mathrm{m^{-3}}
\simeq1.1\times10^{18}\,\mathrm{cm^{-3}}.}
$$
The associated <plasma> wavelength is about $31$ micrometres, illustrating why a short driver structure is required. The given <length> and <energy> gain do not uniquely fix <density> without an assumption about wake amplitude relative to the <plasma> scale. Driver shape, beam loading, relativistic wave-breaking corrections and matching to a <Positron> witness are substantive additional issues; the estimate is a scale calculation, not a completed accelerator design.
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