Question 332495
This requires knowing the equation of the volume and surface area of rectangles

volume: (area of base)*height, 
since base is a square and if its side is "S" then
{{{volume = s^2*H}}}

Surface area:  2* (area of the base) + (perimeter of the base) * height 
{{{Surface Area =2*S^2 + 4*S*H}}} 
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Solve the volume equation for H:   {{{H=Volume/(S^2)}}}
substitute this into the surface area: {{{2*S^2+4*S*H = 2*S^2 + 4*S(Volume/(S^2))}}}

Surface area= {{{SA=2*S^2 + 4*volume/S = 2*S^2 + 4*(216)/S = 2*S^2+864/S}}}
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Take the derivative of SA: (SA)' ={{{ 4*S -864/(S^2)}}}
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set this derivative =0 and solve for S
{{{4S-864/(S^2) = 0}}}
{{{4S=864/(S^2)}}}
{{{4S^3=864}}}
{{{S^3=216}}}
S=6
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Check to see if this a minimum or maximum by taking the second derivative of SA
(SA)" ={{{ 4-864*(-2)/(S^3)=4+3456/(S^3)}}} evaluated at S=6
(SA)"= {{{4+3456/(6^3)=20}}}, since its positive this means, the function curves upward, thus S=6 is a minimum
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Now substitute S=6 into the SA equation to find the lowest surface area
{{{SA= 2*S^2+864/S = 2*6^2+864/6=2*36+144=216}}}