$$ \text { weight }(w)=\frac{4}{3} \pi\left(\frac{\mathrm{D}^3-\mathrm{d}^3}{8}\right) \sigma g $$ Buoyant force $\left(\mathrm{F}_{\mathrm{b}}\right)=1 \times \frac{4}{3} \pi\left(\frac{\mathrm{D}^3}{8}\right) \cdot \mathrm{g}$ For Just Float $\Rightarrow \mathrm{w}=\mathrm{F}_{\mathrm{b}}$ $$ \begin{aligned} & \Rightarrow\left(D^3-d^3\right) \sigma=D^3 \\ & \Rightarrow 1-\frac{d^3}{D^3}=\frac{1}{\sigma} \\ & \Rightarrow 1-\frac{1}{\sigma}=\left(\frac{d}{D}\right)^3 \\ & \Rightarrow\left(\frac{\sigma}{\sigma-1}\right)^{\frac{1}{3}}=\left(\frac{D}{d}\right) \end{aligned} $$