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x  side of the square base   equal to 127,20 in

h height of the cube             equal to  127,20 in

Step-by-step explanation:

Volume of cube Vc is   514500 in³

Let call x side of the square base, then

Area of the base  Ab = x²

Area of (4) lateral sides  Al  =  4 * x*h     h height of the cube

Area of (4) inside partitions Ap = 4*x*h

And Vc =  x²* h    ⇒  514500 = x²*h       ⇒  h   =  514500/ x²

Total area;  At = Ab + 4 Al + 4 Ap

as Al = Ap

At = Ab + 8 Al

At = x²  +  8*x*h               and   h = 2058000

Then area as function of x

A(x) = x² + 8*x* 514500/x²   ⇒        A(x) = x² +  4116000/x

Taking derivative on both sides of the equation

A´(x) = 2x - 4116000/x²       ⇒     A´(x) = 0

2x  - 4116000/x²  = 0

2*x³ - 4116000 = 0

x³  -  2058000 = 0

x³ = 2058000

x  = 127,20 in

And   h  =  2058000/x²        ⇒  h = 2058000/ 16179,84

h = 127,20 in