a cylindrical can with a top and a bottom is to have a volume of 128 cubic centimeters.

Question

a cylindrical can with a top and a bottom is to have a volume of 128 cubic centimeters. find the dimensions that will minimize the amount of material required to construct the can.

Answer 1

V=pi r^2 h.....128 = pi r^2 h, so h = 128/pi r^2 .

 

SA = 2 pi r h + 2 pi r^2 . you can now substitute h

SA = 2 pi r (128/pi r^2 ) + 2 pi r^2 . Since you now have a function in one variable, you can simplify it and take its derivative.

SA = 256r^(-1) + 2 pi r^2 

SA' =-256r^(-2) + 4 pi r. Setting it equal to 0 and solving you get

 

-256/r^2 + 4 pi r = 0

-256 + 4 pi r ^3 = 0

pi r ^3  = 64

r = 4/cube root of pi....approximately 2.7311

h = 128/ pi (4/cube root of pi)^2.......approximately 5.4622.