As you know, the volume of a cylinder is 

V = , 

where pi = 3.14, r is the radius and h is the height.


In your case the volume is fixed:

 =   cm^3.       (1)


The surface area of a cylinder is 

S = ,    (2)

and they ask you to find minimum of (2) under the restriction (1).


Using (1), I can rewrite (2) in the form

S(r) =  +  =  +  =  + .   (3)


The plot below shows the function S(r) =  + , and you can clearly see that it has the minimum.



    

        Plot y = 


To find the minimum, use Calculus: differentiate the function to get

S'(r) =  +  = 

and equate it to zero.


S'(r) = 0   leads you to equation   = ,   which gives 

r =  = 4 cm.


Answer.  r = 4 cm, h =  = 8 cm gives the minimum of the surface area.

A company that manufactures dog food wishes to pack in closed cylindrical tins.
What should be the dimensions of each tin if it is to have a volume of
128πcm³ and the minimum possible surface area:

In the UK, you say "tin". In the US, we say "can". LOL 

The volume of a cylinder is 

 

Which simplifies to 
 

The surface area of a cylinder with a top and bottom is


Substituting for h:



Differentiating w/r r:

Setting the derivative equal to zero:

Dividing through by -4π

Multiplying through by r2




So the radius of each can should be 4 cm. (diameter = 8 cm.)
Substituting in





And the height of each can should be 8 cm.

So the mid-cross section of each can should be an 8 cm x 8 cm square!





Edwin