Question 167653
a large soup can is to be designed so that the can will hold 16π in^3 (about 28oz) of soup. find what height and radius of the cylindrical can will minimize the amount of metal needed.
:
Find the relationship between the radius and height using the V = {{{pi*r^2*h}}}
{{{pi*r^2*h = 16pi}}}
:
h =  {{{((16pi))/((pi*r^2))}}}
Cancel pi
h = {{{16/(r^2)}}}
:
The amt of metal used in the can is determined by the surface area
S.A = {{{(2*pi*r^2) + (2*pi*r*h)}}}
Substitute {{{16/(r^2)}}} for h
S.A = {{{(2*pi*r^2) + (2*pi*r*(16/r^2))}}}
Cancel r:
S.A = {{{(2*pi*r^2) + (2pi(16/r))}}}
Or
S.A = {{{(2*pi*r^2) + ((32/r)pi)}}}
:
Put this in your graphing calc
y = {{{(2*pi*x^2) + ((32/x)pi)}}};   x = radius, y = surface area
Scale it: x: -2, +4; y: -30, +120
;
you graph should look like this:
{{{ graph( 300, 200, -2, 4, -30, 120, (6.28*x^2)+(100.53/x)) }}}
:
You can estimate that minimum surface area about r = 2 inches
Use the minimum feature on your calc: bracket the minimum and it will
 give the exact radius and surface area; 

I got radius (x) = 2; giving a Minimum surface area (y) = 75.398 sq/inches
;
Did this make some sense to you?