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Question Number 5319 by Rasheed Soomro last updated on 07/May/16

$$\mathrm{A}\mathbf{sphere}\mathrm{of}\mathrm{radius}\mathbf{r}\mathrm{contains}\mathrm{a}\mathbf{cube}$$$$\mathrm{inside}\mathrm{it}.\mathrm{All}\mathrm{the}\mathbf{vertices}\mathrm{of}\mathrm{the}\mathrm{cube}$$$$\mathrm{touch}\mathrm{the}\mathbf{surface}\mathrm{of}\mathrm{sphere}.$$$$\mathrm{What}\mathrm{is}\mathrm{the}\mathbf{volume}\mathrm{of}\mathrm{the}\mathrm{cube}?$$

Answered by Rasheed Soomro last updated on 08/May/16

Commented by Rasheed Soomro last updated on 08/May/16

$${\mathrm{Let}}^{\prime }\mathrm{s}\mathrm{cut}\mathrm{the}‘‘\mathrm{sphered}{\mathrm{cube}}^{″}\mathrm{in}\mathrm{two}$$$$\mathrm{equal}\mathrm{parts}\mathrm{in}\mathrm{such}\mathrm{a}\mathrm{manner}\mathrm{that}\mathrm{two}$$$$\mathrm{dimensions}\mathrm{of}\mathrm{the}\mathrm{cube}\mathrm{are}\mathrm{preserved}.$$$$\mathrm{Above}\mathrm{is}\mathrm{the}\mathrm{cross}-\mathrm{section}\mathrm{of}\mathrm{one}\mathrm{of}$$$$\mathrm{the}\mathrm{two}\mathrm{parts}.\mathbf{r}\mathrm{is}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{sphere}$$$$\mathrm{and}\mathbf{s}\mathrm{is}\mathrm{measure}\mathrm{of}\mathrm{the}\mathrm{side}\mathrm{of}\mathrm{the}\mathrm{cube}.$$$$\mathrm{clearly},$$$$\mathrm{s}=\sqrt{{\mathrm{r}}^2+{\mathrm{r}}^2}=\sqrt{2}\mathrm{r}$$$$\mathrm{volume}\mathrm{v}\mathrm{of}\mathrm{the}\mathrm{cube}$$$$\mathrm{v}={\mathrm{s}}^3={\left(\sqrt{2}\mathrm{r}\right)}^3=2\sqrt{2}{\mathrm{r}}^3$$

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