Consider a solid sphere of radius R and mass density  $\rho \left(r\right)={\rho }_o(1-\frac{r^2}{R^2})$,   $0<r\le R$. The minimum density of a liquid in which it will float is 

In case of minimum density of liquid, sphere will be floating while completely submerged will be floating while completely submerged, so  $mg=U$

$$\begin{gathered} \Rightarrow mg=\left(\underset{0}{\overset{R}{\int }}{\rho }^{\left(4\pi r^{2}dr\right)}\right)g=U \\ {\rho }_{o}g\underset{0}{\overset{R}{\int }}(1-\frac{r^2}{R^2})4\pi r^{2}dr=\frac{4}{3}\pi R^3{\rho }_{1g} \Rightarrow {\rho }_1=\frac{2{\rho }_o}{5} \end{gathered}$$