Question 716288
The volume has a height and a circumference.  The circular side has the same height and circumference as the volume.  

{{{pi*r^2*h=27*pi}}} and {{{2*pi*r*h=18*pi}}}.  The area equation is like stretching the circular sheet out straight, which was the circumference, and then the other dimension would be height.  Circumference times height is the 18 cm^2 area.


Those are two equations with two unknowns of r and h.  Solve the equations as a system.


LET ME TRY ANALYZING A SOLUTION AGAIN BUT DIFFERENTLY.

r is radius, h is height.


The volume is given and for this cylinder shaped can, volume is {{{pi*r^2*h=27*pi}}} cubic centimeters.


What about the curved circular side?  If you cut it from top to bottom and flatten it into a rectangle, then you see the top to bottom height, h, and now the circumference, {{{2*pi*r}}}, is the "length".  This area was given in the problem description and is {{{2*pi*r*h=18*pi}}} cubic centimeters.  The r and the h are the same as for the volume.


The system of equations to solve for r and h is this:
{{{highlight(pi*r^2*h=27*pi)}}}
{{{highlight(2*pi*r*h=18*pi)}}}


The equations really should be simplified before working with them to solve the system.  Both equations have a factor of {{{pi}}} on their left and right sides.  No need! For both equations, divide both sides by pi.  Also, in the area equation, you should divide both sides by 2.   You do those things and you obtain:


SIMPLIFIED SYSTEM
{{{highlight(r^2*h=27)}}}
{{{highlight(r*h=9)}}}
The route to a finished solution should now be very simple.