Question

the schwarzschild radius of a black hole with a mass of 1.0 $m_{sun}$ is 3.0 km. what is the schwarzschild radius of a black hole with a mass of 10.0 $m_{sun}$? 15.0 km 3.0 km 30.0 km 45.0 km

Explanation:

Step1: Identify the proportionality

The Schwarzschild radius $R_s$ is proportional to the mass $M$ of the black - hole. That is, $R_s\propto M$.

Step2: Set up the proportion

If $R_{s1}$ is the radius for mass $M_1$ and $R_{s2}$ is the radius for mass $M_2$, then $\frac{R_{s1}}{M_1}=\frac{R_{s2}}{M_2}$. We know that $M_1 = 1.0M_{\text{Sun}}$, $R_{s1}=3.0\text{ km}$, and $M_2 = 10.0M_{\text{Sun}}$.

Step3: Solve for $R_{s2}$

Cross - multiply the proportion: $R_{s2}=\frac{M_2}{M_1}\times R_{s1}$. Substitute the values: $R_{s2}=\frac{10.0M_{\text{Sun}}}{1.0M_{\text{Sun}}}\times3.0\text{ km}=30.0\text{ km}$.

Answer:

C. 30.0 km