Expert: Ahmed Salami - 12/18/2008

Question
A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volune of the solid is 12 cubic centimeters. Find the radius of the cylinder that produces the  minimum surface area.

Answer
Hi John,
The volume of the solid is that of a sphere plus a cylinder. If the length of the cylinder is h, the volume V is
V = (4/3)#r^3 + #r^2.h
from which
h = [V - (4/3)#r^3] / #r^2
 = V/#r^2 - (4/3)r
The surface area of the solid is that of the sphere and the cylinder
A = 4#r^2 + 2#rh
 = 4#r^2 + 2#r[V/#r^2 - (4/3)r]
 = 4#r^2 + 2V/r - (8/3)#r^2
 = (4/3)#r^2 + 2V/r
The minimum area occurs when dA/dr = 0, but
dA/dr = (8/3)#r - 2V/r^2
equating to zero,
(8/3)#r - 2V/r^2 = 0
(8/3)#r = 2V/r^2
r^3 = 3V/4#

There you go!

Regards