## Ratio of Densities of Planet Core and Mantle Problem

Suppose a planet of mass\[M\]

and radius \[R\]

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 | \[\frac{0.6M}{\frac{4}{3} \pi (0.3R)^3}\] | \[\frac{0.4M}{\frac{4}{3} \pi \times 0.973R^3}\] |

\[\frac{0.6M}{\frac{4}{3} \pi (0.3R)^3}\frac{0.4M}{\frac{4}{3} \pi \times 0.973R^3}\]

This simplifies to

\[\frac{200}{9} : \frac{400}{973}\]

.