## Ratio of Densities of Planet Core and Mantle Problem

Suppose a planet of mass
$M$
$R$
consists of two layers of material, each with a constant density. The core contains 60% of the planets mass, but has radius equal to 30% of the planets radius. The mantle constitutes the rest of the mass.
What is the ratio Density of Core: Density of Mantle?
We can draw up the table below.
 Core Mantle Mass $0.6M$ $0.4M$ Volume $\frac{4}{3} \pi (0.3R)^3$ $\frac{4}{3} \pi R^3 - \frac{4}{3} \pi (0.3)^3 = \frac{4}{3} \pi \times 0.973R^3$ Density $\frac{0.6M}{\frac{4}{3} \pi (0.3R)^3}$ $\frac{0.4M}{\frac{4}{3} \pi \times 0.973R^3}$
The required ratio is then
$\frac{0.6M}{\frac{4}{3} \pi (0.3R)^3}\frac{0.4M}{\frac{4}{3} \pi \times 0.973R^3}$

This simplifies to
$\frac{200}{9} : \frac{400}{973}$
.