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?

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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.

... 9. Using the First and Second Derivative tests as appropriate ... To find critical points, put h'(x) = 0. ... at x = -2 and its value is -1 and local maximum value at x ...

...Find the first order condition for a critical point of this ... a maximum or a minimum or an inflection point? ... minimum we check the sign of the second derivative: ...

... Note: it is also possible to combine this technique with the first and second derivative tests to find which critical points are local minima and ...

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... to find all critical points and use the second derivative to find all inflections ... Use a graph to identify each critical point as a local maximum, a local ...

... Let us first find the critical points for the given function for that find the first ... Therefore by second derivative test Q = 50 will be maximum point of the ...

... 3. Find dy/dx by implicit differentiation. ... 4. Determine the critical numbers for the given function and classify each critical point as a relative maximum...