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I have no idea how to do this problem at all.

A cylindrical can without a top is made to contain V cm^3 of liquid. Find the dimensions that will minimize the cost of the metal to make the can.

Since no specific volume is given the smallest amount of metal for the can would be zero, which would held zero cm^3 of liquid. How is this wrong? It is not possible to make a cylinder out of a negative amount of metal.

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    $V$ is given, although not numerically. Write equations for the volume and area of the cylinder in terms of the radius and height. Solve for those variables, using $V$ as a constant.2012-04-03
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    But I don't understand how 0 isn't the answer.2012-04-03
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    Because a cylinder with zero area has zero volume. So unless $V=0$, the answer cannot be $0$.2012-04-03
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    The problem doesn't specify that the volume is > 0.2012-04-03
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    @Jordan: EXACTLY. The problem doesn't specify what $V$ is! You are trying to specify that it is zero. Why not let go of that faulty assumption and proceed with the calculus? It's not a trick question...2012-04-03
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    Does the problem ask you to show or state why the solution you find is a global minimum? Have you been shown the second derivative test, what it means for a function to be convex or concave, and the use of the number line of the first derivative to evaluate critical points? Usually, this material is also covered, so that you don't just blindly go solving for roots of the derivative without knowing what result it really gives you.2012-04-03
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    Yes I know about concavity and the first and second derivative tests.2012-04-03
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    Dear Jordan, You are (or at least were) confused about the point of the question. The point is: imagine you want to make cans which hold say $400$ cm${}^3$ of liquid as cheaply as possible (i.e. using as little metal as possible), then how should you make them: tall and thin, short and fat, somewhere in between? The point of the question is to work this out, and not just for $400$ cm${}^3$, but for an arbitrary volume $V$. It's not good telling someone who wants to make and sell $400$ cm${}^3$ volume cans of soup that they should make cans that hold *no* soup. You have to tell them how ...2012-04-04
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    ... they can most cheaply make the cans of the size they want. Regards,2012-04-04

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