Chapter 12.5, Problem 12WE

To determine

To find: The volume of the smaller cube.

Answer to Problem 12WE

The volume of the smaller cube is $\frac{8}{3\sqrt{3}}{\text{cm}}^3$ .

Explanation of Solution

Given information:

The diagonal of one cube is $2\text{cm}$ .

The diagonal of another cube is $4\sqrt{3}\text{cm}$ .

The volume of larger cube is $64{\text{cm}}^3$ .

Calculation:

The volume is proportional to the cube of side and diagonal is proportional to the side so volume is proportional to the cube diagonal.

  $\begin{array}{c}\frac{\text{Total}\text{volume}\text{of}\text{larger}\text{cube}}{\text{Total}\text{volume}\text{of}\text{smaller}\text{cube}}={\left(\frac{4\sqrt{3}}{2}\right)}^3\\ \frac{\text{64}}{\text{Total}\text{volume}\text{of}\text{smaller}\text{cube}}={\left(2\sqrt{3}\right)}^3\\ \text{Total}\text{volume}\text{of}\text{smaller}\text{cube}=\frac{64}{8{\left(\sqrt{3}\right)}^3}\\ =\frac{8}{3\sqrt{3}}{\text{cm}}^3\end{array}$

Therefore,the volume of the smaller cube is $\frac{8}{3\sqrt{3}}{\text{cm}}^3$ .