A manufacturer produces cardboard boxes that are open at the top and sealed at the base. The base is rectangular and its length is double its width. Let x denote the width in metres. the surface area of each such box is fixed to be 3 square metres. The manufacturer wishes to determine the height h and the base width x, in metres, of the box so that its volume is as large as possible.
a) Express the volume V in terms of x and h
b) Show that 2x^2+6xh =3 (consider the surface area)
c) Express h in terms of x and deduce that
d) Use calculus to find the value of x so that V is as large as possible. Justify your answer. What is the largest possible value of the volume?
The maximum volume is found using a critical point. The solution is detailed.