If a cube has surface area ^@S^@ and volume ^@V^@, then find the volume of the cube of surface area ^@3S^@.
Answer:
^@3 \sqrt{ 3 }V^@
- Let the edge of the cube be ^@a^@. Then,
Surface area, ^@S = 6a^2^@, and
Volume, ^@V = a^3^@. - We can say that ^@a = \left(\dfrac{ S } {6} \right)^{ 1 \over 2 }^@. Now let us put this value of ^@a^@ in the expression for volume. ^@V^@ then becomes:
^@\left(\dfrac{ S } {6}\right)^{ 3\over 2 }^@. - Thus, for another cube with surface area ^@3S^@, the volume will be:
^@\begin{align} &\left(\dfrac{ 3S } {6}\right)^{ 3 \over 2 } \\ = & ( 3 )^{ 3 \over 2 } \times \left(\dfrac{ S } {6}\right)^{ 3\over 2 } \\ = & ( 3 )^{ 3 \over 2 }V \\ = & 3 \sqrt{ 3 }V \end{align}^@ - Hence the volume of the other cube is ^@3 \sqrt{ 3 }V^@.