United Parcel Service has contracted you to design an open box with a square base that has a volume of 5000 cubic inches. Express the surface area S of the box as a function of x.

Answer :

The surface area of the box as a function of x will be f(x) = x² + 20000/x² where x is the length of the square base's edge.

Here it is given that the box has a square base.

This implies that length = breadth

Now, the volume of a box = lbh = 5000 cu in

where l is the length of the box. b is the breadth of the box, and h is the height of the box.

or, x²h = 5000

or, h = 5000/x²

Let the surface area function be f(x) where x is the length of the edge of the square base.

Since the box is open, the surface area will be

lb + 2bh + 2lh

= x² + 10000/x² + 10000/x

= x² + 20000/x²

Hence the surface area as a function of x will be

f(x) = x² + 20000/x²

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