calculus problem with optimization (i think)?

Question

determine the radius and height of the cylinder of maximum volume that can be obtained by revolving a rectangle of perimeter 24 inches about one of its sides.

how do you solve that??

thanks.

Answer 1

let H be the height of the cylinder. And r the radius

the perimeter is P = 4r+2h =24 this gives h = 12-2r (1)

the volume is v = Pi r^2 *h replace h by 12-2r Pi =3.1416

V = Pi r^2 * (12-2r) = Pi * (12r^2 -2r^3)= 2*Pi (6r^2-r^3)

to find the maximum derive dV/dr =2 *Pi (12 r -4r^2)= 8*Pi *r (3-r)

you have a maximum for r =3 and the height using 1 is 6


V = Pi *9*6 =54 Pi