Chapter 12.1, Problem 38WE

To determine

To find: the volume when the diagonals are $4\sqrt{3}\text{cm}$ .

Answer to Problem 38WE

The volume is $\frac{8\sqrt{3}}{9}{\text{cm}}^3$ when the diagonals are $4\sqrt{3}\text{cm}$ .

Explanation of Solution

Given information:

A diagonal of a cube joins two vertices not in the same face and the diagonals are $4\sqrt{3}\text{cm}$ long.

Calculation: since, A diagonal of a cube joins two vertices not in the same face,

Thus, the factor can be find as:

  $\begin{array}{l}k=\frac{4\sqrt{3}}{2}\\ k=2\sqrt{3}\text{cm [divide]}\end{array}$

Now, the ratio of the larger cube and smaller cube is cube of $k$ factor,

  $\begin{array}{l}\frac{v_l}{v_s}=k^3\\ \frac{{\left(4\sqrt{3}\right)}^2}{v_s}={\left(2\sqrt{3}\right)}^3\text{ [simplify]}\\ \frac{64}{v_s}=24\sqrt{3}\text{ [multiply]}\\ v_s=\frac{64}{24\sqrt{3}}\text{ [cross multiplication]}\\ v_s=\frac{8\sqrt{3}}{9}\text{ [simplify]}\end{array}$

Thus, The volume is $\frac{8\sqrt{3}}{9}{\text{cm}}^3$ when the diagonals are $4\sqrt{3}\text{cm}$ .