A square sheet of metal of side a has squares cut on all its corners and the sides bent to form an open box. What is the maximum volume of the box?

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The sheet of metal we have has sides equal to a. Let the sides of the squares that are cut from all the corners be x.

Now we have a length equal to a – 2x, to fold on all the sides and create the box.

The volume of the box is (a – 2x)^2*x = (a^2 + 4x^2 – 4ax)*x

V = a^2x + 4x^3 – 4ax^2

To maximize V, we...

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