Question

If V denotes the volume of a cube and T its surface area, show that: $\mathrm{V}=\frac{\mathrm{T}}{36}\sqrt{6\mathrm{T}}$.

Answer 1

Dear Student,
Here is the solution of your asked query:
$$\begin{gathered} \mathrm{Let},\mathrm{a}\mathrm{be}\mathrm{each}\mathrm{edge}\mathrm{of}\mathrm{cube} \\ \mathrm{Then},\mathrm{volume}\mathrm{of}\mathrm{cube},\mathrm{V}={\mathrm{a}}^3...\left(\mathrm{i}\right) \\ \mathrm{Surface}\mathrm{area},\mathrm{T}=6{\mathrm{a}}^2...\left(\mathrm{ii}\right) \\ \mathrm{R}.\mathrm{H}.\mathrm{S}.=\frac{\mathrm{T}}{36}\sqrt{6\mathrm{T}} \\ =\frac{6{\mathrm{a}}^2}{36}\sqrt{6\times 6{\mathrm{a}}^2}\{\mathrm{using}(\mathrm{ii}\left)\right\} \\ =\frac{{\mathrm{a}}^2}{6}\times 6\mathrm{a} \\ ={\mathrm{a}}^3=\mathrm{V}\{\mathrm{using}(\mathrm{i}\left)\right\} \\ \mathrm{So},\mathrm{L}.\mathrm{H}.\mathrm{S}.=\mathrm{R}.\mathrm{H}.\mathrm{S}. \end{gathered}$$
Regards