First, let's recognize that we must find the dimensions of a cylinder that yields a volume of 500 cm^3 and has the least surface area possible. This means that we must minimize the quantity 2pir^2 + 2pirh.
Since the volume of a cylinder is pir^2h, we have
pir^2h = 500.
h = 500/(pir^2).
Plugging back into our equation for surface area yields
A(r) = 2pir^2 + 1000/r.
Since we want to find the minimum of this function, we must take the derivative with respect to r and set the derivative equal to 0.
0 = 4pir + -1000/r^2.
Now, all we have to do to solve for r. Rearranging yields
1000/r^2 = 4pir
1000 = 4pir^3
r^3 = 250/pi
r = root(3)(250/pi).
Solving for h with this value of r yields
h = 500/(piroot(3)(250/pi)^2
h = 500/(pi(250^(2/3)/pi^(2/3)))
h = (2times250^(3/3))/(250^(2/3)times(pi^(3/3)/pi^(2/3)))
h = 2times250^(1/3)/pi^(1/3)
h = 2root(3)(250/pi).
So our final answer is a cylinder with a radius of 4.3 cm and height of 8.6 cm.