How Do You Calculate the Density of Earth's Core Given Its Mantle Density?

[mendem03]

Homework Statement

Assuming the Earth to be comprised of only a uniform density mantle and a uniform density
core and the mean density of the mantle to be 4700 kg=m3, determine the mean density of
the Earth's core. Take Earth's radius to be 6400 km and the radius of the core to be 3500 km.
Assume the Earth's mean density is 5520 kg=m3

Homework Equations

density = mass/volume

The Attempt at a Solution

V_c = 1.80 * 10^20 meter cubed
V_e = 1.10 * 10^21 meter cubed
V_m = V_e - V_c
= 9.20 * 10^20

M_e = M_c + M_m
M_e = D_c*V_c + D_m*V_m
D_c = (M_e - D_m*V_m)/V_c
D_c = 9222.22

I think I got the wrong answer so can someone help me with it.

 

[Smapple]

From the givens, I got
D_c = (M_e - M_m) / V_c
= [ D_e*V_e - ((D_m (V_e - V_c)) ] / V_c

which is equivalent to yours and an answer, though slightly different, of the same order of magnitude.