  We theoretically study the static and dynamic behavior of a BEC immersed in a
large Fermi sea of ultracold atoms under conditions of tunable interspecies
interaction. The degenerate Bose-Fermi mixture is kept in an elongated trap,
typical for a single-beam optical dipole trap. We focus on the case of
repulsive Bose-Fermi interaction and develop mean-field models to simulate the
system over a wide range of repulsion strength. We further get analytical
solutions in the regimes of phase separation and weak interaction. We obtain
static density profiles and the frequency of the radial breathing mode, which
is an elementary dynamic phenomenon of the mixture. Our results unveil the
structure of the Bose-Fermi interface and describe the origin of the frequency
shift of the breathing mode when the components become phase-separated at
strong repulsion. We show that the mediated interaction between bosons induced
by the Fermi sea can be understood as an adiabatic second-order mean-field
effect, which is valid also beyond the weak-interaction regime for relevant
experimental conditions. These results are consistent with our recent
observations in a mixture of $^{41}$K and $^6$Li.
