Find the values of r which minimizes the surface area.TIn cans come in various shapes and sizes, but what factors influence their design? In...

We know that the surface area of the cylinder is given by
:

SA = 2pir^2 + 2pi*r*h

But we
know that the volume is :

v = r^2 * pi *
h

 

Given that v is a fixed
value.

Then we will write h as a function of
r.

==> h = v/r^2 *
pi

Now we will substitute into
SA.

==> SA = 2pi*r^2 + 2pi*r *(v/r^2
pi)

==> SA = 2pi*r^2 + 2v/
r

Now we will
differentiate:

==> (SA)' = 4pir -
2v/r^2

Now we will find critical
values.

==> 4pi*r - 2v/r^2 =
0

==> 2v/r^2 =
4pi*r

==> Multiply by
r^2

==> 2v =
4pi*r^3

==> r^3 =
v/2pi

==> r =
(v/2pi)^1/3

Then
the value of r that given minimum surface area is r=
(v/2pi)^1/3