An open rectangular cardboard box with a square base is to have a volume of 256cm^3. find the dimensions of the box if the area of cardboard used is as small as possible

Let
x> the length side of the square base of the box
y> the height of the box

we know that
volume of the box=b²*h
b=x
h=y
volume=256 cm³
so
256=x²*y>y=256/x²>  equation 1

The amount of material used is directly proportional to the surface area, so we will minimize the amount of material by minimizing the surface area.

surface area of the cardboard=area of the base+perimeter of base*height
area of the base=x²
perimeter of the base=4*x
height=y
surface area=x²+4x*y> equation 2
substitute equation 1 in equation 2
SA=x²+4x*[256/x²]> SA=x²+1024/x

step 1 
find the first derivative of SA and equate to zero
2x+1024*(-1)/x²=0> 2x=1024/x²> x³=512> x=8 cm
y=256/x²> y=256/8²> y=4 cm

the answer is
the length side of the square base of the box is 8 cm
the height of the box is 4 cm

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