Acardboard box without a lid is to have a volume of 5,324 cm3. find the dimensions that minimize the amount of cardboard used. (let x, y, and z be the dimensions of the cardboard box.)

The area of the box is

which we want to minimize subject to the constraint .

The Lagrangian is

with critical points where the partial derivatives are 0:

Notice that

Substituting the latter into either or will end up suggesting that is infinite, so we throw out this case.

If , then

We ignore the case where because that would make the volume 0. Then

so we have one critical point at

which give a minimum area of about 1829.41 sq. cm.

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step-by-step explanation: