Final answer:
The Schwarzschild radius of a 100 million-solar-mass black hole is 2.96 × 10⁹ meters, or approximately 3,000 kilometers, which corresponds to option C) 3,000 km.
Step-by-step explanation:
The Schwarzschild radius (Rs) of a black hole is calculated using the formula Rs = (2GM) / c², where G is the gravitational constant (6.67 × 10⁻¹¹ N·m²/kg²), M is the mass of the black hole, and c is the speed of light (3 × 10⁸ m/s). To find the Schwarzschild radius of a black hole with a mass of 100 million solar masses, we use the given mass of the Sun (2 × 10³⁰ kg) to find the mass of the black hole (M = 100 million × 2 × 10³⁰ kg = 2 × 10³⁷ kg). Substituting these values into the formula:
Rs = 2 × (6.67 × 10⁻¹¹ N·m²/kg²) × (2 × 10³⁷ kg) / (3 × 10⁸ m/s)² = 2 × (6.67 × 10⁻¹¹) × (2 × 10³⁷) / (9 × 10¹⁶) m = 2 × (6.67 × 10⁻¹¹ × 2 × 10³⁷) / (9 × 10¹⁶) m = 2.96 × 10⁹ m.
Hence, the Schwarzschild radius of a 100 million-solar-mass black hole is 2.96 × 10⁹ meters, or approximately 3,000 kilometers (C). This means the correct answer is option C) 3,000 km.