Question concerning average density

[Emethyst]

Homework Statement

A planet or radius R has a core of radius R/3 and density 2d, a middle layer from radius R/3 to 2R/3 and density d/2, and an outer shell from radius 2R/3 to R and a density of d. What is the average density as a whole?

Homework Equations

d=m/v

The Attempt at a Solution

Not sure where else to stick this question so I put in the basic math section here (if it's the wrong section please do move it). I figured that to solve this question I would need to simple add all 3 density values given together and divide by 3. This seems too easy, however, and I'm not sure if I can treat the mass as a constant. Would this be the right method or am I missing something here? If anyone could be of assistance it would be greatly appreciated, thanks in advance.

 

[LCKurtz]

Figure out the mass of each portion using m = dv then divide the total mass by the total volume to get the average density.

 

[HallsofIvy]

No, adding all 3 density will not give you the "average" density. That would work only if all densities were for an equal volume. You need to use the basic definition of "density". Find the mass of the entire sphere and divide by its volume.

Multiply the density of each region by its volume (that is LCKurtz's formula). The first is easy- its volume the volume of a sphere of radius R/3. The second is trickier. Find the volume of a sphere of radius 2R/3 and then subtract the volume of a sphere of radius R/3. For the outer shell, find the volume of a sphere of radius R and subtract the volume of a sphere of radius 3R/3.