Q39. - Geometry

Question

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

Step-by-Step Solution

Answer

The length of the edges is 6 cm.

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 is x,

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,

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

 

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