Answer:
[tex]\large \boxed{6}[/tex]
Step-by-step explanation:
The formula for the volume of a right triangular prism is
V = ½ach, where
a = the height of the base
c = the length of a side of the base, and
h = the height of the prism
In your prism, a = h, so
V = ½ch²
The base is an equilateral, so
[tex]\begin{array}{rcl}\dfrac{h}{c} & = & \dfrac{\sqrt{3}}{2}\\\\h & = & \dfrac{c\sqrt{3}}{2}\\\end{array}[/tex]
Then
[tex]\begin{array}{rcl}V & = & \dfrac{1}{2}ch^{2}\\\\81 & = & \dfrac{1}{2}\times c \times \left(\dfrac{c\sqrt{3}}{2} \right )^{2}\\\\81 & = & \dfrac{1}{2}\times c^{3}\times\dfrac{3}{4}\\\\81 & = & \dfrac{3}{8}c^{3}\\\\c^{3} & = &\dfrac{81 \times 8}{3}\\\\ & = & 27 \times 8\\c & = & 3 \times 2\\c & = & \mathbf{6}\\ \end{array}\\\text{The lengths of the sides of the base are $\large \boxed{\mathbf{6}}$}[/tex]
