Optimize a Hexagonal Prism of volume 500cm^3?

Question

requires minimal surface area

Answer 1

The area is minimized with a regular hexagon cross section.

Area = 2x(area of hexagon) + 6x(side of hexagon)x(height)

Volume = (Area of hexagon)x(height)

(Area of hexagon) = 3sqrt(3)s^2

0.5 = 3hsqrt(3)s^2

h = 0.5/{3sqrt(3)s^2}

Area = 2x(3sqrt(3)s^2) + 6x(s)x(0.5/{3sqrt(3)s^2})

=6sqrt(3)s^2 + 1/[sqrt(3)s]

dA/sd = 12sqrt(3)s - 1/[sqrt(3)(s^2)] = 0

12sqrt(3)s = 1/[sqrt(3)(s^2)]

36s = 1/s^2

s^3 = 1/36

s = 1/(36)^(1/3) = 0.302853 cm

h= 1.049115 cm

Area = 2.859553 cm^2