(b) \(\sqrt6 : 1\)
Let the length of each side of the equilateral triangle be a cm.
Also, let the length of each side of the regular hexagon be b cm.
Area of equilateral triangle = \(\frac{\sqrt3}{4}a^2\)
Area of regular hexagon = \(6\times\frac{\sqrt3}{4}b^2=\frac{3\sqrt3}{2}b^2\)
Since, volume of prism with equilateral triangle base = Volume of prism with hexagonal base
⇒ \(\frac{\sqrt3}{4}a^2\times{h}=\frac{3\sqrt3}{2}b^2\times{h}\)
⇒ \(\frac{a^2}{b^2}=\frac{3\sqrt2}{2}÷\frac{\sqrt3}{4}=\frac61\) ⇒ a : b = \(\sqrt6 : 1\).