Radius of core, using densities

[bocobuff]

Homework Statement

A planet with R=4000km and $$\rho$$_avg=5g/cm³. The planet is made of quartz all the way to the core, which is made of iron. The densities of quart and iron are $$\rho$$_quartz=2.65g/cm³ and $$\rho$$_iron=7.874g/cm³. Calculate the radius of the core.

Homework Equations

$$\rho$$=M/V
V_sphere=4/3$$\pi$$r³

The Attempt at a Solution

I solved for M_planet=$$\rho$$_avg=5g/cm³*4/3$$\pi$$4000km and got 1.34*10²⁷g. Then with assuming the M_planet=M_{Tot. Qrtz}+M_{Tot. Iron}, I tried using various substitutions of my unknowns and I can't get anywhere close to being able to solve for any of them. I know it's some simple calculus, but I am lost and don't remember how to do this. The biggest problem for me is that the radius is cubed and I forgot what to do to solve for that.
Any fresh ideas would help a lot.
Thanks

 

[Carid]

The volume of the planet is equal to the volume of the core plus the volume of the mantle.
That's one equation. Then you have your mass equation.
From these two you can substitute for the volume of the mantle to find the volume of the core and thus arrive at core radius.

 

[bocobuff]

So after a bunch of substitutions and algebra solving for V_core, I got
V_core=($$\rho$$_avg*V$$\rho$$_planet - $$\rho$$_mantle*V_planet) / ($$\rho$$_core - $$\rho$$_mantle) and got 1.206*10²⁶g/cm³.
Solving for the radius I got
r=$$\sqrt[3]{V_core/(4pi/3)}$$ and got 3.0665*10⁸cm which is 3065km, roughly 75% of the planet's r. Sound right?

 

[Carid]

No I don't think so...

VolumePlanet = VolumeMantle + VolumeCore

now

VolumeMantle = VolumePlanet - VolumeCore

so substituing

VolumePlanet = (VolumePlanet - VolumeCore) + VolumeCore

Now why did we do that? Because if we now multiply the volumes by the densities to get the masses, things are much more interesting...

VolumePlanet * DensityAverage= ((VolumePlanet - VolumeCore) * DensityMantle) + (VolumeCore * DensityCore)

Now we know all those numbers except VolumeCore. We are on our way...

I've probably given you too much help but I couldn't see any other way.