Thickness of mantle

burnout_128

1. The problem statement, all variables and given/known data

Given the density of the mantle to be 4700 kg/m^3, the density of the core to be 9700 kg/m^3, the radius of the planet to be 6050 km and the mean density to be 5240 kg/m^3, find the thickness of the mantle.

2. Relevant equations

I tried using the formula M = D * V and V = (4/3)(pie)(r^3) but there seems to be too much missing data.

mgb_phys

You have an equation linking volume (and so mass) of a sphere with radius.
Now write the equation for the volume of the shell (ie mantle) which will depend on the thickness of the mantle ie, on the radius of the core.

Then you have an equation something like
overall average mass = mass of core (r_core) + mass mantle ( r_core )

burnout_128

So I end up having three equations...
1) M_total = V_total * D_total
2) M_core = V_core * D_core
3) M_mantle = V_mantle * D_mantle

1) I have the V because I am given the radius and can also figure out the M because I am given the D.
2 & 3) I only have both D's and nothing else.

I also know that: 4) M_total = M_core + M_mantle
Similarly, I know that: 5) V_total = V_core + V_mantle

Equations (4) and (5) seems useless as I don't have information about the mass of either the mantle or the core and don't have information about the volume of the core or mantle.

Is there some way I can substitute equations... or am I even going about it the right way?

mgb_phys

You are on the right path,
Write the volumes in terms of radius of Earth and radius of Core, note the volume of core and mantle are related.
Then you can write equation 4 in terms of this core radius.