Minimize the suface area of an open box (no lid) with a square base that had a volume of 5000 cubic inches by expressin suface area S of the box as a function of x. It is given that the each side of the sqaure base is 10 inches.
To minimize the total surface area of the box for a given volume at least two of the dimensions, length, width or height should be such that they can be varied.
The problem gives the total volume of the box as 5000 cubic inches and it is also given that it has a square base with the length of each side equal to 10 inches. There is no variable left here than can be varied.
As a result it is neither possible to express the surface area in terms of a variable x nor is it possible to minimize the surface area.
Let length of the side of the base be x inches, and hight of the box be
Volume of the box= area of the base . hight of the box
`S=x^2+2xx(x xx h+x xx h)`
x=10 inches ,and h=10 inches
so surface area of the box will fixed no minimum or maximum.