Surface area S of the box as a function?

Question

An open rectangular box (no top) with volume 5 cubic meters has a square base. Express the surface area S of the box as a function of the length x of one of the sides.

Answer 1

Let x = length of the sides of the base, and h = the height.  The volume is given as 5 m³:

 

Volume = area of base * height

5 = x²*h

 

So 5/x² = h

 

The surface area, S, of the box is:

 

S = (Area of Base) + 4*(Area of a Vertical Side)

S = x² + 4*(xh)

 

Substitute 5/x² in place of h:

 

S = x² + 4*x*(5/x²)

S = x² + 20/x