Calculating Planet Mantle Density

The mean density ($\rho$) of the planet is the total mass divided by the total volume. The planet consists of a core and a mantle. We need to find the density of the mantle ($\rho_m$).

Planet Composition and Volumes

The planet has a core of radius $r_c$ and a mantle extending to radius $r$. The total volume ($V$) of the planet is the volume of a sphere with radius $r$, and the core volume ($V_c$) is the volume of a sphere with radius $r_c$. The mantle volume ($V_m$) is the difference between the total volume and the core volume.

• Total Volume: $V = \frac{4}{3}\pi r^3$
• Core Volume: $V_c = \frac{4}{3}\pi r_c^3$
• Mantle Volume: $V_m = V - V_c = \frac{4}{3}\pi r^3 - \frac{4}{3}\pi r_c^3 = \frac{4}{3}\pi (r^3 - r_c^3)$

Mass Calculation

The mass of each part is its density multiplied by its volume.

• Mass of Core: $M_c = \rho_c V_c = \rho_c \left(\frac{4}{3}\pi r_c^3\right)$
• Mass of Mantle: $M_m = \rho_m V_m = \rho_m \left(\frac{4}{3}\pi (r^3 - r_c^3)\right)$
• Total Mass: $M = M_c + M_m = \rho_c \left(\frac{4}{3}\pi r_c^3\right) + \rho_m \left(\frac{4}{3}\pi (r^3 - r_c^3)\right)$

Deriving Mantle Density

Using the definition of mean density, $\rho = \frac{M}{V}$, we can set up the equation:

$ \rho = \frac{\rho_c \left(\frac{4}{3}\pi r_c^3\right) + \rho_m \left(\frac{4}{3}\pi (r^3 - r_c^3)\right)}{\frac{4}{3}\pi r^3} $

Cancel the common factor $\frac{4}{3}\pi$:

$ \rho = \frac{\rho_c r_c^3 + \rho_m (r^3 - r_c^3)}{r^3} $

Multiply both sides by $r^3$:

$ \rho r^3 = \rho_c r_c^3 + \rho_m (r^3 - r_c^3) $

Rearrange to solve for $\rho_m (r^3 - r_c^3)$:

$ \rho_m (r^3 - r_c^3) = \rho r^3 - \rho_c r_c^3 $

Finally, isolate $\rho_m$:

$ \rho_m = \frac{\rho r^3 - \rho_c r_c^3}{r^3 - r_c^3} $

Divide the numerator and denominator by $r^3$ to match the option format:

$ \rho_m = \frac{\rho - \rho_c \frac{r_c^3}{r^3}}{1 - \frac{r_c^3}{r^3}} = \frac{\rho - \left(\frac{r_c}{r}\right)^3\rho_c}{1-\left(\frac{r_c}{r}\right)^3} $