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#