A rectangular sheet with perimeter of 33cm and dimensions x by y are rolled into a cylinder, x being the perimeter What values x and y give the largest volume?

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Answer 1 · 2016-09-23T11:53:21

$22$ [cm] and $11$ [cm]

$x+y=L$
$V = pi r^2 h$ where

$x = 2pi r$ and $h = y$ so

$V = pi(x/(2pi))^2y = 1/(4pi)x^2y$

but $y = L-x$ so

$V = 1/(4pi)x^2(L-x)$

The extremal points obey

$(dV)/(dx) = ((2 L - 3 x) x)/(4pi)=0$

so the stationary points are

$x = 0$ no volume

$x = (2L)/3$ max volume

so $x = 2 xx 33/3=22$

and

$y = 33-22=11$