1. [tex]1.69\cdot 10^{11} m[/tex]
The Schwarzschild radius of an object of mass M is given by:
[tex]r_s = \frac{2GM}{c^2}[/tex] (1)
where
G is the gravitational constant
M is the mass of the object
c is the speed of light
The black hole in the problem has a mass of
[tex]M=5.7\cdot 10^7 M_s[/tex]
where
[tex]M_s = 2.0\cdot 10^{30} kg[/tex] is the solar mass. Substituting,
[tex]M=(5.7\cdot 10^7)(2\cdot 10^{30}kg)=1.14\cdot 10^{38} kg[/tex]
and substituting into eq.(1), we find the Schwarzschild radius of this black hole:
[tex]r_s = \frac{2(6.67\cdot 10^{-11})(1.14\cdot 10^{38} kg)}{(3\cdot 10^8 m/s)^2}=1.69\cdot 10^{11} m[/tex]
2) 242.8 solar radii
We are asked to find the radius of the black hole in units of the solar radius.
The solar radius is
[tex]r_S = 6.96\cdot 10^5 km = 6.96\cdot 10^8 m[/tex]
Therefore, the Schwarzschild radius of the black hole in solar radius units is
[tex]r=\frac{1.69\cdot 10^{11} m}{6.96\cdot 10^8 m}=242.8[/tex]
