Answer:
(I). The Schwarzschild radius is ![2.94\times10^{8}\ km](https://tex.z-dn.net/?f=2.94%5Ctimes10%5E%7B8%7D%5C%20km)
(II). The Schwarzschild radius is 17.7 km.
(III). The Schwarzschild radius is ![1.1\times10^{-7}\ km](https://tex.z-dn.net/?f=1.1%5Ctimes10%5E%7B-7%7D%5C%20km)
(IV). The Schwarzschild radius is ![7.4\times10^{-29}\ km](https://tex.z-dn.net/?f=7.4%5Ctimes10%5E%7B-29%7D%5C%20km)
Explanation:
Given that,
Mass of black hole ![m= 1\times10^{8} M_{sun}](https://tex.z-dn.net/?f=m%3D%201%5Ctimes10%5E%7B8%7D%20M_%7Bsun%7D)
(I). We need to calculate the Schwarzschild radius
Using formula of radius
![R_{g}=\dfrac{2MG}{c^2}](https://tex.z-dn.net/?f=R_%7Bg%7D%3D%5Cdfrac%7B2MG%7D%7Bc%5E2%7D)
Where, G = gravitational constant
M = mass
c = speed of light
Put the value into the formula
![R_{g}=\dfrac{2\times6.67\times10^{-11}\times1\times10^{8}\times1.989\times10^{30}}{(3\times10^{8})^2}](https://tex.z-dn.net/?f=R_%7Bg%7D%3D%5Cdfrac%7B2%5Ctimes6.67%5Ctimes10%5E%7B-11%7D%5Ctimes1%5Ctimes10%5E%7B8%7D%5Ctimes1.989%5Ctimes10%5E%7B30%7D%7D%7B%283%5Ctimes10%5E%7B8%7D%29%5E2%7D)
![R_{g}=2.94\times10^{8}\ km](https://tex.z-dn.net/?f=R_%7Bg%7D%3D2.94%5Ctimes10%5E%7B8%7D%5C%20km)
(II). Mass of block hole ![m= 6 M_{sun}](https://tex.z-dn.net/?f=m%3D%206%20M_%7Bsun%7D)
We need to calculate the Schwarzschild radius
Using formula of radius
![R_{g}=\dfrac{2MG}{c^2}](https://tex.z-dn.net/?f=R_%7Bg%7D%3D%5Cdfrac%7B2MG%7D%7Bc%5E2%7D)
Put the value into the formula
![R_{g}=\dfrac{2\times6.67\times10^{-11}\times6\times1.989\times10^{30}}{(3\times10^{8})^2}](https://tex.z-dn.net/?f=R_%7Bg%7D%3D%5Cdfrac%7B2%5Ctimes6.67%5Ctimes10%5E%7B-11%7D%5Ctimes6%5Ctimes1.989%5Ctimes10%5E%7B30%7D%7D%7B%283%5Ctimes10%5E%7B8%7D%29%5E2%7D)
![R_{g}=17.7\ km](https://tex.z-dn.net/?f=R_%7Bg%7D%3D17.7%5C%20km)
(III). Mass of block hole m= mass of moon
We need to calculate the Schwarzschild radius
Using formula of radius
![R_{g}=\dfrac{2MG}{c^2}](https://tex.z-dn.net/?f=R_%7Bg%7D%3D%5Cdfrac%7B2MG%7D%7Bc%5E2%7D)
Put the value into the formula
![R_{g}=\dfrac{2\times6.67\times10^{-11}\times7.35\times10^{22}}{(3\times10^{8})^2}](https://tex.z-dn.net/?f=R_%7Bg%7D%3D%5Cdfrac%7B2%5Ctimes6.67%5Ctimes10%5E%7B-11%7D%5Ctimes7.35%5Ctimes10%5E%7B22%7D%7D%7B%283%5Ctimes10%5E%7B8%7D%29%5E2%7D)
![R_{g}=1.1\times10^{-7}\ km](https://tex.z-dn.net/?f=R_%7Bg%7D%3D1.1%5Ctimes10%5E%7B-7%7D%5C%20km)
(IV). Mass = 50 kg
We need to calculate the Schwarzschild radius
Using formula of radius
![R_{g}=\dfrac{2MG}{c^2}](https://tex.z-dn.net/?f=R_%7Bg%7D%3D%5Cdfrac%7B2MG%7D%7Bc%5E2%7D)
Put the value into the formula
![R_{g}=\dfrac{2\times6.67\times10^{-11}\times50}{(3\times10^{8})^2}](https://tex.z-dn.net/?f=R_%7Bg%7D%3D%5Cdfrac%7B2%5Ctimes6.67%5Ctimes10%5E%7B-11%7D%5Ctimes50%7D%7B%283%5Ctimes10%5E%7B8%7D%29%5E2%7D)
![R_{g}=7.4\times10^{-29}\ km](https://tex.z-dn.net/?f=R_%7Bg%7D%3D7.4%5Ctimes10%5E%7B-29%7D%5C%20km)
Hence, (I). The Schwarzschild radius is ![2.94\times10^{8}\ km](https://tex.z-dn.net/?f=2.94%5Ctimes10%5E%7B8%7D%5C%20km)
(II). The Schwarzschild radius is 17.7 km.
(III). The Schwarzschild radius is ![1.1\times10^{-7}\ km](https://tex.z-dn.net/?f=1.1%5Ctimes10%5E%7B-7%7D%5C%20km)
(IV). The Schwarzschild radius is ![7.4\times10^{-29}\ km](https://tex.z-dn.net/?f=7.4%5Ctimes10%5E%7B-29%7D%5C%20km)