Draw a cylindrical container (with a lid), so as to contain $1$ liter of water, using a minimal amount of metal.
Could you give me some hints how we could do that??
Do we have to use the Lagrange multipliers method??
2026-04-04 05:17:10.1775279830
Do we have to use the Lagrange multipliers method?
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So, we have $V=\pi r^2 h=1$ and wish to minimize the surface area $S=2\pi r^2+2\pi rh$.
HINT:
Note that we have $V=\pi r^2 h=1$ as a constraint.
So, eliminate $h$ as $h=1/(\pi r^2)$.
Then, substitute this expression for $h$ into the expression for $S=2\pi r^2 + 2\pi r h$, take the derivative with respect to $r$, set this equal to zero and find the value of $r$ for the minimum of $S$ (i.e, the least amount of metal).