Chapter 12.1, Problem 39WE

To determine

To find: the length of the edges when the volume is $54\sqrt{3}{\text{cm}}^3$ .

Answer to Problem 39WE

The length of the edges is $6\text{cm}$ when the volume is $54\sqrt{3}{\text{cm}}^3$ .

Explanation of Solution

Given information:

All nine edges of a right triangular prism are congruent.

Calculation: since, nine edges of a right triangular prism are congruent.

To find the edges length, let edges length is $x$ ,

So the base area can be,

  $b=\frac{\sqrt{3}}{4}{x}^2$

Thus, the volume is,

  $\begin{array}{l}v=bh\\ 54\sqrt{3}=\frac{\sqrt{3}}{4}{x}^2\left(x\right)\text{ [substitute]}\\ 54\sqrt{3}=\frac{\sqrt{3}}{4}{x}^3\text{ [mutliply]}\\ x^3=216\text{ [simplify]}\\ x=6\text{ [cube root]}\end{array}$

Thus, The length of the edges is $6\text{cm}$ when the volume is $54\sqrt{3}{\text{cm}}^3$ .