# Differentiation : Critical Point - Find Maximum Value

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

V=x-(2/3)x^3

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?

https://brainmass.com/math/derivatives/differentiation-critical-point-maximum-value-18905

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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 ...

#### Solution Summary

The maximum volume is found using a critical point. The solution is detailed.