Answer:
[tex]\frac{R_1^3 * \rho _1}{R_2^3 * \rho _2}[/tex]
Explanation:
The mass density of an object with uniform mass distribution its defined as
[tex]\rho = \frac{mass}{volume}[/tex].
So, if we know the volume, and, we can obtain the mass of the object:
[tex]mass = volume * \rho[/tex].
Now, we can take the planets as spheres, of course, this is only an approximation, but good enough for us. The volume of a sphere of radius r its:
[tex]Volume_{sphere} = \frac{4}{3} \pi r^3[/tex]
So, for our planets, the mass its given by:
[tex]mass = volume * \rho\\mass = \frac{4}{3} \pi r ^3 \rho[/tex],
so
[tex]mass_{planet1} = \frac{4}{3} \pi R_1^3 \rho_1[/tex]
[tex]mass_{planet2} = \frac{4}{3} \pi R_2^3 \rho_2[/tex]
Now, we can take the ratio:
[tex] \frac{mass_{planet1}}{mass_{planet2}} = \frac{ \frac{4}{3} \pi R_1 ^3 \rho_1 }{ \frac{4}{3} \pi R_2 ^3 \rho_2 } [/tex]
Now, we can just cancel the [tex] \frac{4}{3} \pi [\tex] that appear in both sides of our fractions, and finally obtain:
[tex]\frac{R_1^3 * \rho _1}{R_2^3 * \rho _2}[/tex].
