Question

All nine edges of right triangular prism are congruent. Find the length of these edges if the volume is$54\sqrt{3}{\text{\hspace{0.17em}\hspace{0.17em}cm}}^3$.

Short answer

The length of the edges is6 cm.

Step-by-step solution

Step 1. Given information.

All nine edges of a right triangular prism are congruent.

Step 2. Write the concept.

Since nine edges of a right triangular prism are congruent.

To find the edges length, let edges length isx,

So, the base area is:

$b=\frac{\sqrt{3}}{4}{x}^2$

Step 3. Determine the volume.

Now, the volume is:

$\begin{array}{c}v=bh\\ 54\sqrt{3}=\frac{\sqrt{3}}{4}{x}^2\left(x\right)\text{[substitute]}\\ 54\sqrt{3}=\frac{\sqrt{3}}{4}{x}^3\text{[mutliply]}\end{array}$

So,

$\begin{array}{c}{x}^3=216\text{[simplify]}\\ x=6\text{[cuberoot]}\end{array}$

Then, the length of the edges is 6 cm when the volume is$54\sqrt{3}{\text{\hspace{0.17em}\hspace{0.17em}cm}}^3$ .