  We study an evaporating black hole in the boundary conformal field theory
(BCFT) model under the fully time-dependent AdS-Vaidya spacetime geometry. We
introduce the time-dependent finite bath termed the effective Hawking radiation
region. This is described by a nontrivial BCFT solution that acts as a
time-dependent brane which we call the moving end-of-the-radiation (METR) brane
that leads to a new type of Hubeny-Rangamani-Takayanagi surface. We further
examine the island formulation in this particular time-dependent spacetime. The
Page curve is calculated by using Holographic Entanglement Entropy (HEE) in the
context of double holography.
