Is the Schwarzschild radius a material surface or a radius scale for the event horizon?

Find the Schwarzschild radius of a mass \(6.0\times10^{30}\,\mathrm{kg}\). Use \(G=6.67\times10^{-11}\,\mathrm{N\,m^2\,kg^{-2}}\) and \(c=3.00\times10^8\,\mathrm{m\,s^{-1}}\).

A compact object has radius \(20\,\mathrm{km}\) and Schwarzschild radius \(35\,\mathrm{km}\). In this model, is it inside its horizon condition?

Find the escape speed from radius \(5.0\times10^7\,\mathrm{m}\) around a mass \(4.0\times10^{30}\,\mathrm{kg}\).

What mass has Schwarzschild radius \(30\,\mathrm{km}\)? Use \(G=6.67\times10^{-11}\,\mathrm{N\,m^2\,kg^{-2}}\) and \(c=3.00\times10^8\,\mathrm{m\,s^{-1}}\).

At what radius, in terms of \(r_s\), does the Newtonian escape-speed formula give \(v_{\mathrm{esc}}=c/2\)?

A star's Schwarzschild radius is \(9.0\,\mathrm{km}\), but its actual radius is \(9000\,\mathrm{km}\). By what factor would its radius need to shrink to reach \(r_s\)?

A compact body has radius \(3r_s\). Use the escape-speed model to find \(v_{\mathrm{esc}}\) as a fraction of \(c\).

A body of mass \(M\) and initial radius \(R\) is compressed without changing its mass. Derive the minimum compression factor \(k\), where final radius is \(R/k\), needed for \(R/k\le r_s\).

Treat \(r_s\) as the radius scale of a black hole and define an average density \(\bar\rho=3M/(4\pi r_s^3)\). Derive \(\bar\rho(M)\) and interpret how it changes as \(M\) increases.