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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 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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The length of the diagonal of a rectangular billboard.
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Linear Transformations, Rotations, Submodules and Subspaces, (7) Let F=R, let V=R^2 and let T be the linear transformation from V to V which is rotation clockwise about the origin by pi-radians. Show that every subspace of V is an F[X]-submodule f..
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