Question

Under some circumstances, a star can collapse into an extremely dense object made mostly of neutrons and called a neutron star. The density of a neutron star is roughly 10¹⁴ times as great as that of ordinary solid matter. Suppose we represent the star as a uniform, solid, rigid sphere, both before and after the collapse. The star's initial radius was 8.0×10⁵ km (comparable to our sun); its final radius is 18 km. Assume that its mass does not change during the collapse. What is the ratio of the final volume of the neutron star to the initial volume of the star?

1) 10¹⁴
2) 10¹⁵
3) 10¹⁶
4) 10¹⁷

Answer 1

To find the ratio of the final volume of the neutron star to the initial volume of the star, we need to calculate the volume of each using the given radius values and then compute the ratio. The ratio is approximately 1.63 x 10¹⁵.

Step-by-step explanation:

To find the ratio of the final volume of the neutron star to the initial volume of the star, we can use the formula for the volume of a sphere:

V = (4/3)πr³

where V is the volume and r is the radius.

For the initial star, the radius is given as 8.0×10⁵ km. Plugging this value into the formula, we get:

V_initial = (4/3)π(8.0×10⁵)³

For the final neutron star, the radius is given as 18 km. Plugging this value into the formula, we get:

V_final = (4/3)π(18)³

Now we can calculate the ratio of the final volume to the initial volume:

Ratio = V_final / V_initial

Simplifying the equation and canceling out common terms, we get:

Ratio = (18)³ / (8.0×10⁵)³

Using a calculator, we find that the ratio is approximately 1.63 x 10¹⁵.