Question 117363
Basically you have to find the relation between the radius and height of a right circular cylinder with given volume for minimum surface area.


Cans are hollow cylinders with both ends closed.
So, total surface area is {{{S = 2*pi*r^2 + 2*pi*r*h}}}.
{{{S = 2*pi*r(r + h)}}}

The volume is {{{V = pi*r^2*h}}}
{{{h = V/(pi*r^2)}}}


So, {{{S = 2*pi*r(r + V/(pi*r^2))}}} [substituting the value of 'h']
{{{S = 2pi*r^2 + 2V/r}}}


For minimum surface area,
{{{dS/dr = 0}}}
{{{4*pi*r - 2V/r^2 = 0}}}
{{{r = root(3,V/(2*pi))}}}


So, the height is {{{h = V/(pi*(root(3,V/(2*pi)))^2)}}}
{{{h = V/(pi*root(3,V^2/(4*pi^2)))}}}
{{{h = root(3,4V/pi)}}}


a) Substituting V = 500 cc,
{{{r = root(3,500/(2*pi)) = 4.30}}} cm
{{{h = root(3,4*500/pi) = 8.60}}} cm


b) {{{S = 2*pi*4.3(4.3 + 8.6) = 348.53}}} sq cm
So, cost of material per can = $(0.2 x 348.53) = $69.7.