What dimensions of the rectangle will result in a cylinder of maximum volume if you consider a rectangle of perimeter 12 inches in which it forms a cylinder by revolving this rectangle about one of its edges?

1 Answer
Feb 5, 2015

A cylinder's volume, #V=pir^2h#
The width of the rectangle, which forms the circumference, #w=2pir#
graph{x+y=12 [-2.49, 25.99, -1.51, 12.74]}
The graph shows possible lengths for the sides of the rectangle if they must add up to 12 inches (#w# is horizontal and #h# is the vertical axis).
At what point along that curve is #w^2/(4pi^2)*h=r^2h# the greatest?

graph{(12x-x^2)/(4pi) [-0.89, 13.35, -1.027, 6.1]}

Since #r=w/(2pi)=(12-h)/(2pi)#, you can plot a graph of the volume for different values of #h# with the equation #y=pi*(12-x)/(4pi^2)*x=(12x-x^2)/(4pi)#. Differentiating to find the maximum:

#dy/dx=12/(4pi)-(2x)/(4pi)=3/pi-x/(2pi)=0#

#x=h=6#

Since #h=6#, #w# must also equal #6#

#V=pir^2*h=pi*(6/(2pi))^2*6=533pi=1674#