  We present a simple model for an evolving radio galaxy, as it expands into
the hot, X-ray emitting interstellar medium (ISM) of its host galaxy or
similarly into the hot intracluster medium (ICM) of its host cluster. We solve
the governing equations numerically on a grid of model parameters in order to
present simple analytical tools for X-ray observations of the shocked shell
that is pushed out by the cocoon and the associated cavity in the cluster
emission. We apply these tools to the well known example of Perseus A to show
that its time-averaged kinetic luminosity probably exceeds 10^46 ergs/sec, much
larger than the estimated current power. We show how future observations can be
used to extract useful source parameters such as the average kinetic power and
the source age, and discuss detectability of sources at various stages of their
lives.
