= Solution
For a <bound-cloud light echo>, the relevant observed delay is the \i[excess] source–cloud–observer path over the direct source–observer path, with the interval dilated by cosmological expansion. For a cloud displaced by $s$ toward us,
$$
\boxed{\Delta t_{\rm obs}=(1+z)\frac{\sqrt{R_\perp^2+s^2}-s}{c}}.
$$
The formula assumes source and cloud separation is small compared with their cosmological distance, so their distant observer directions are effectively parallel. For the transverse configuration $s=0$,
$$
\boxed{\Delta t_{\rm obs}=3\frac{87.6\ \mathrm{kpc}}c\simeq8.57\times10^5\ \mathrm{yr}}.
$$
The local quasar-to-cloud travel time is about $2.86\times10^5$ yr; the factor three is needed for the interval measured by us. A foreground cloud shortens the echo delay and a background cloud lengthens it. A unique value beyond this transverse estimate needs the otherwise unspecified depth, even if the cloud's intrinsic response is instantaneous.
\Image[/past-exam-of-the-astrophysics-course-of-the-university-of-oxford/2015-bound-cloud-light-echo.png]
{title=Projected separation and unknown cloud depth determine the excess path and observed echo delay}
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