This is using calculus

Ok, so it is always a good idea to draw a diagram when we can. see the diagram below.

let the length of each side of the base of the box be x

let the height of the box be y

then the volume is given by:

V = length*width*height = x^2 * y

we want the volume to be 442368

so we have x^2 * y = 442368

we want everything in terms of x, so let's solve for y

=> y = 442368/(x^2)

Now the surface area is the area of all the faces (the top not included since it's an open top). so the surface area is given by:

S = x^2 + xy + xy + xy + xy = x^2 + 4xy

=> S = x^2 + 4x(442368/(x^2))

=> S = x^2 + 1769472/x

we want S to be a minimum.

for minimum we find S' and set it to zero:

S' = 2x -1769472x^-2

for min, set S' = 0

=> 2x - 1769472x^-2 = 0

=> 2x^3 - 1769472 = 0

=> 2x^3 = 1769472

=> x^3 = 1769472/2 = 884736

=> x = cuberoot(884736)

=> x = 96

so the base of the box must be 96 for minimum surface area. what must the height be?

y = 442368/(x^2) = 442368/(96^2) = 48

so the dimensions of the box for minimum surface area are:

width = length = 96 and height = 48