A white dwarf has a density of approximately \(10^{9}\) kilograms per cubic meter \(\left(\mathrm{kg} / \mathrm{m}^{3}\right) .\) Earth has an average density of \(5,500 \mathrm{kg} / \mathrm{m}^{3}\) and a diameter of \(12,700 \mathrm{km} .\) If Earth were compressed to the same density as a white dwarf, how large would it be?

To understand how the volume of Earth is calculated, let's start by acknowledging that Earth is roughly a sphere. This means we can use the formula for the volume of a sphere, which is:
\ \[ V = \frac{4}{3} \pi r^3 \] Here, \( V \) represents the volume, and \( r \) is the radius of Earth. Given that Earth's diameter is 12,700 km, we can find the radius by dividing the diameter by 2, resulting in 6,350 km or 6,350,000 meters.
Using the formula and plugging in the radius, we calculate the volume of Earth. Remember to keep your units consistent. This means converting kilometers to meters if necessary and using the right constants.

Once the volume of Earth is calculated, determining its mass becomes straightforward using its average density. Density is defined as mass per unit volume, expressed as:
\ \[ \rho = \frac{m}{V} \] Here, \( \rho \) is the density, \( m \) is the mass, and \( V \) is the volume. To calculate Earth's mass, we rearrange the formula to:
\ \[ m = \rho \times V \] Earth’s average density, given as 5,500 kg/m³, can then be multiplied by its calculated volume to find the mass. This gives us the total mass of Earth based on its volume and density.

Understanding the sphere volume formula is crucial when we need to determine how compressing Earth to the density of a white dwarf changes its size. The sphere volume formula is:
\ \[ V = \frac{4}{3} \pi r^3 \] By knowing the mass of Earth and the density of a white dwarf, which is given as \(10^9\) kg/m³, we use the density formula in reverse to find the new volume:
\ \[ V = \frac{m}{\rho} \] We then recalculate Earth's volume based on this new density. Lastly, to find the new radius, we rearrange the sphere volume formula to isolate \( r \):
\ \[ r = \left( \frac{3V}{4\pi} \right)^{\frac{1}{3}} \] Finally, multiplying the new radius by 2 gives us the new diameter of Earth when compressed to the density of a white dwarf.