A hollow spherical shell at outer radius R floats just submerged...

In equilibrium $mg = {F}_{B}$

$\dfrac{4}{3}\pi\left({R}^{3} - {r}^{3}\right){ρ}_{0}g = \dfrac{4}{3}\pi{R}^{3}{ρ}_{w}g$

\left[1 - {\left(\dfrac{r}{R}\right)}^{3}\right]\dfrac{27}{8}{ρ}_{w} = {ρ}_{w}$</p><p>$1 - \dfrac{{r}^{3}}{{R}^{3}} = \dfrac{9}{27}$</p><p>$1 - \dfrac{1}{3} = \dfrac{{r}^{3}}{{R}^{3}}$</p><p>$\dfrac{2}{3} = \dfrac{{r}^{3}}{{R}^{3}}$</p><p>$\dfrac{r}{R} = {\left(\dfrac{2}{3}\right)}^{1 / 3}$</p><p>$1 - \dfrac{{r}^{3}}{{R}^{3}} = \dfrac{8}{27}$</p><p>$\dfrac{{r}^{3}}{{R}^{3}} = 1 - \dfrac{8}{27} = \dfrac{19}{27}$</p><p>$r = 0 . 89$