Calculation of true thickness of the bed

Question: The radius of the planet is 2500km and its mean density is 3.3 gm/cc. If the planet has a core with a radius 1000 km and the uniform density of the overlying mantle is 3.0 gm/cc. what is the density of the core?

(Thanks to Chandrasekhar, AKNU)

Solution:    

                     

Given that

      Planet radius(R) = 2500km

      Mean density ($\rho$)= 3.3 gm/cc

      Core radius $r_c$=1000 km

      Density of the mantle $\rho_m$=3.0 gm/cc

Then Density of the core $\rho_c$=?

 Formula:

$\rho R^{3}=\rho_{c}r_{c}^{3}+\rho_{m}(R^{3}-r_{c}^{3})$

 By substituting the given values in the above formulae, we get

$\Rightarrow (3.3) (2500)^{3}=\rho_{c}(1000)^{3}+(3.0)((2500)-(1000)^{3})$

$\Rightarrow 5.166\times 10^{10}=\rho_{c}10^{9}+4.3875\times10^{10}$

$\Rightarrow \rho_{c}10^{9}=7.785\times 10^{9}$

$\Rightarrow \rho_{c}=7.785 gm/cc$