A prism with a regular hexagonal base and a height of 3 in. has a volume of 288 square roots of 3 cu in. what is the length of each edge of the base?

Question

callisto · Answer

Volume = Base area x height
$$288\sqrt3 = base \ area \times 3$$$$96\sqrt3 = base \ area $$|dw:1336438625789:dw|For a regular hexagon, it is made up of 6 equi. triangles.
Height of each small triangle 
----------------------- = tan 60
(x/2)
Height of each small triangle = (x tan60) /2
Area of each smaller triangle = x [ (x tan60) /2 ] /2 
                                              = x^2 (sqrt3) /4
$$Area \ of \ hexagon = 6 \times [ x^2 (\sqrt3) /4 ]   =  96 \sqrt3$$$$3x^2 (\sqrt3)    =  192 \sqrt3$$$$x^2 = 64$$
x =8 or x=-8 (rejected)