volume of a sphere = (4/3)πr³, where r is the radius. The mass of the Sun is 2 × 10³⁰ kg. The density of water is 1000 kg/m³. The Centaurus galaxy has a supermassive black hole at its centre. The mass of the black hole is 5.5 × 10⁷ times the mass of the Sun, and its density is 1000 times that of water. Assuming that the black hole is spherical, use the information given to calculate its diameter in metres. Give your answer in standard form to 3 s.f.
To find the diameter of the black hole, we first need to calculate its mass and then its volume using the density formula. After that, we can use the volume of a sphere formula to find the radius and subsequently the diameter.
1. Calculate the mass of the black hole:
[
text{Mass of black hole} = 5.5 times 10^7 times text{Mass of the Sun} = 5.5 times 10^7 times 2 times 10^{30} text{ kg}
]
[
= 1.1 times 10^{38} text{ kg}
]
2. Calculate the density of the black hole:
[
text{Density of black hole} = 1000 times text{Density of water} = 1000 times 1000 text{ kg/m}^3 = 10^6 text{ kg/m}^3
]
3. Calculate the volume of the black hole:
Using the formula for density,
[
text{Density} = frac{text{Mass}}{text{Volume}} implies text{Volume} = frac{text{Mass}}{text{Density}} = frac{1.1 times 10^{38}}{10^6}