Question

A spherical shell with internal radius r is converted into a solid sphere whose volume is equal to its cavity. Find the thickness of the shell.

Answer 1

Dear Student,

$$\begin{gathered} LetRbetheexternalradius. \\ Volumeofsphere=Volumeofshell \\ Volumeofcavity=Volumeofshell \\ \frac{4}{3}{\mathrm{\pi r}}^3=\frac{4}{3}\mathrm{\pi }\left({\mathrm{R}}^3-{\mathrm{r}}^3\right) \\ \Rightarrow {\mathrm{r}}^3={\mathrm{R}}^3-{\mathrm{r}}^3 \\ \Rightarrow 2{\mathrm{r}}^3={\mathrm{R}}^3 \\ \Rightarrow \mathrm{R}=2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\mathrm{r} \\ \mathrm{Now}\mathrm{thickness}\mathrm{of}\mathrm{shell}=\mathrm{R}-\mathrm{r}=2^{\raisebox{1ex}{${\displaystyle 1}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle 3}$}\right.}\mathrm{r}-\mathrm{r}=\left(2^{\raisebox{1ex}{${\displaystyle 1}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle 3}$}\right.}-1\right)\mathrm{r} \end{gathered}$$

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