Radius of core, using densities

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To calculate the radius of a planet's core made of iron, given its average density and total radius, the mass equations for the planet and its components must be established. The volume of the planet is the sum of the volumes of the core and the mantle, leading to a relationship between their densities and volumes. By substituting the volume of the mantle into the mass equation, one can isolate the volume of the core. The radius of the core can then be determined using the formula for the volume of a sphere. The final calculated core radius is approximately 3065 km, which raises questions about its proportion relative to the planet's total radius.
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Homework Statement


A planet with R=4000km and \rhoavg=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 \rhoquartz=2.65g/cm3 and \rhoiron=7.874g/cm3. Calculate the radius of the core.


Homework Equations


\rho=M/V
Vsphere=4/3\pir3


The Attempt at a Solution


I solved for Mplanet=\rhoavg=5g/cm3*4/3\pi4000km and got 1.34*1027g. Then with assuming the Mplanet=MTot. Qrtz+MTot. 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
 
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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.
 
So after a bunch of substitutions and algebra solving for Vcore, I got
Vcore=(\rhoavg*V\rhoplanet - \rhomantle*Vplanet) / (\rhocore - \rhomantle) and got 1.206*1026g/cm3.
Solving for the radius I got
r=\sqrt[3]{V<sub>core</sub>/(4pi/3)} and got 3.0665*108cm which is 3065km, roughly 75% of the planet's r. Sound right?
 
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.
 
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