In 1999, scientists discovered a new class of black holes with masses 100 to 10,000 times the mass of our sun, but occupying less space than our moon. Suppose that one of these black holes has a mass of 1000 suns and a radius equal to one-half the radius of our moon. What is the density of the black hole in g/cm³? The radius of our sun is 6.955 x 10⁵ km and it has an average density of 1.410 x 10³ kg/m³. The diameter of the moon is 2.16 x 10³ miles.

Solution:

1) The mass of our sun:

6.955 x 10⁵ km = 6.955 x 10⁸ m

V = (4/3)πr³

V = (4/3) (3.14159) (6.955 x 10⁸ m)³

V = 1.40922275 x 10²⁷ m³

mass of one sun = (1.40922275 x 10²⁷ m³) (1.410 x 10³ kg/m³) = 1.987004 x 10³⁰ kg = 1.987004 x 10³³ g

mass of 1000 suns = 1.987004 x 10³⁶ g

2) volume of black hole:

radius of moon = 2.16 x 10³ mi / 2 = 1.08 x 10³ mi

radius of black hole = 1.08 x 10³ mi / 2 = 540 mi

Use Google calculator to change 540 miles to cm = 8.690 x 10⁷ cm

V = (4/3)πr³

V = (4/3) (3.14159) (8.690 x 10⁷ cm)³

V = 2.748828 x 10²⁴ cm³

3) calculate density of black hole:

1.987004 x 10³⁶ g / 2.748828 x 10²⁴ cm³ = 7.23 x 10¹¹ g/cm³