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A clump of matter does not need to be extraordinarily dense to have an escape velocity greater than the speed of light, as long as its mass is large enough. Use the formula for the Schwarzschild radius [tex]R_{s}[/tex] to calculate the volume [tex]\frac{4}{3} \pi R_s^3[/tex] inside the event horizon of a black hole of mass [tex]M[/tex].

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Certainly! Let's go through the step-by-step solution to determine the volume inside the event horizon of a black hole with a given mass [tex]\( M \)[/tex].

### Step 1: Understand the Inputs and Constants

1. Gravitational Constant ([tex]\( G \)[/tex]): [tex]\( G = 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \)[/tex]
2. Speed of Light ([tex]\( c \)[/tex]): [tex]\( c = 299792458 \, \text{m/s} \)[/tex]
3. Mass of the Sun ([tex]\( M_{\odot} \)[/tex]): [tex]\( M_{\odot} = 1.989 \times 10^{30} \, \text{kg} \)[/tex]
4. Mass of the Black Hole ([tex]\( M \)[/tex]): Let's assume it is 10 times the mass of the Sun. Therefore, [tex]\( M = 10 \, M_{\odot} = 10 \times 1.989 \times 10^{30} \, \text{kg} = 1.989 \times 10^{31} \, \text{kg} \)[/tex].

### Step 2: Calculate the Schwarzschild Radius

The Schwarzschild radius [tex]\( R_s \)[/tex] is given by the formula:

[tex]\[ R_s = \frac{2 G M}{c^2} \][/tex]

Plug in the known values:

[tex]\[ R_s = \frac{2 \times 6.67430 \times 10^{-11} \times 1.989 \times 10^{31}}{(299792458)^2} \][/tex]

After performing the calculation:

[tex]\[ R_s = 29541.27 \, \text{meters} \][/tex]

### Step 3: Calculate the Volume Inside the Event Horizon

The volume [tex]\( V \)[/tex] inside the event horizon is given by the formula for the volume of a sphere:

[tex]\[ V = \frac{4}{3} \pi R_s^3 \][/tex]

Plug in the Schwarzschild radius:

[tex]\[ V = \frac{4}{3} \pi (29541.27)^3 \][/tex]

### Step 4: Perform the Final Calculation

[tex]\[ V = \frac{4}{3} \pi (29541.27)^3 \][/tex]

After performing the calculation:

[tex]\[ V = 1.07988 \times 10^{14} \, \text{m}^3 \][/tex]

### Summary of the Solution
- Gravitational Constant (G): [tex]\(6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \)[/tex]
- Speed of Light (c): [tex]\(299792458 \, \text{m/s} \)[/tex]
- Mass of the Sun ([tex]\( M_{\odot} \)[/tex]): [tex]\(1.989 \times 10^{30} \, \text{kg} \)[/tex]
- Mass of the Black Hole (M): [tex]\(1.989 \times 10^{31} \, \text{kg} \)[/tex]
- Schwarzschild Radius ([tex]\( R_s \)[/tex]): [tex]\(29541.27 \, \text{m} \)[/tex]
- Volume inside the Event Horizon: [tex]\(1.07988 \times 10^{14} \, \text{m}^3 \)[/tex]

This comprehensive calculation determines the volume inside the event horizon of a black hole with a mass 10 times that of the Sun.
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