The base of prism is equal is equilateral triangle. The distance from the center of one base to one of the vertices of the other base is $\iota$ . Then altitude of the prism for which the volume is greatest is

From the figue,   $l^2=h^2+x^2$

Area of base ( equilateral triangle) is  $\frac{\sqrt{3}}{4}{a}^2.$

Also, $\frac{3x}{2}=a\sin {60}^{\circ }=\frac{\sqrt{3}}{2}a$      $\therefore$  $a=\sqrt{3}x$

Now, Volume (V) = Height X Area of base

$=h\frac{\sqrt{3}}{4}{a}^2=h\cdot \frac{\sqrt{3}}{4}\cdot 3\cdot x^2=\frac{3\sqrt{3}}{4}h(l^2-h^2)$

$=\frac{3\sqrt{3}}{4}(h{l}^2-h^3)\Rightarrow \frac{dV}{dh}=0\Rightarrow h=\frac{l}{\sqrt{3}},$

which is point of maxima