How to maximize volume of tube created with a piece of paper of area $1200\ \rm{cm}^2$

Question

I am doing a math project where I have to maximize the volume of a tube created by rolling this piece of paper. The paper has a fixed area of $1200\ \rm{cm}^2$. I have been trying to use optimization to find the volume by using the equation $max=\pi r^2h$ and the restraint equation $2\pi rh=1200$. When I substitute and make the derivative equal to zero, there is only one term and I cannot find any critical points. I am not sure what I'm doing wrong but I have searched everywhere and can't find an answer.

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Answer 1

Use $ 2\pi rh = 1200\ \rm{cm}^2 = A$ and $V = \pi r^2h$. Now $A$ is a constant so write $h = A / (2\pi r)$. Then put $r$ value in $V$. Hence you got a function between $V$ and $r$.

Now $V = Ar/2 $, so $ dV/dr = A/2 $, hence it is an increasing function and as such has no maxima. The larger the radius and the smaller the height of the tube, the more volume it has. ($V$ proportional to $r$, inversely to $h$.)