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Question

An open box with a square base is to be made out of a given quantity of cardboard of area c2 square units. Show that the maximum volume of the box is c363 cubic units.

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Solution

The area of the square piece is given to be c2 square units.
Since the base is a square, let the length and breadth of the resulting box be l and the height be h.
Therefore, l2+4lh=c2, equating the areas.
Also, the volume of the box is thus given by length × breadth × height = l2h
We can write the volume only in terms of l as l2×c2l24l=lc2l34
Differentiating this w.r.t l and equating it to zero, we get c23l2=0
l=c23
The volume thus becomes lc2l34=c23×c2c234
=c3×c26
=c363

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