Question 607004
an open top cylindrical container has a volume of 100 pie. 
construct a single-variable object function suited to minimizing the surface area?
:
x = the radius
The volume of a cylinder: V = {{{pi*x^2*h}}}
{{{pi*x^2*h}}} = {{{100pi}}}
divide both side by pi
x^2*h = 100
h = {{{100/x^2}}}
:
The surface area of a open topped cylinder: S.A. = {{{2*pi*r*h}}} + {{{pi*r^2}}}
replace h with 
S.A. = {{{2*pi*x*(100/x^2)}}} + {{{pi*x^2}}}
Cancel x
S.A. = {{{2*pi*(100/x)}}} + {{{pi*x^2}}}
:
Graph this equation; y = surface area
{{{ graph( 300, 200, -5, 12, -100, 400, x-2, (2*pi*(100/x))+(pi*x^2)) }}}
Looks like minimum surface area occurs when x=4.64 (radius), height = 4.64 also (100/4.64^2)
S.A. ~ 200 sq/units