Cylinder: Ratio Volume/Surface Area

[LURCH]

So a friend of mine was trying to figure out cooling rates for a pot of coffee. He wanted to see how much faster the coffee would cool when he doesn't fill the pot all the way. He is starting from the assumtion that the main determinant for cooling rate will be how the ratio of Surface Area to Volume changes as the fluid level changes. Since the fluid essentially forms a cylinder, and the height is the only real variable dimension, I thought there must be a formula that will yield this ratio, because volume is a function of surface area.

Does anyone know of such a formula? One that will yield the ratio of Volume to Surface Area of a cylinder when the radius remains constant and the height varies?

My thoughts so far:
If one dimension of any 3-D object is increased, the surface area of that object will increase by the square of that dimension, but the volume will increase by the cube of the dimension, right? So won't the volume always be the surface area to the power of three-halves (v=a^3/2)?

Thanks for any help you can render.

 

[slider142]

Since the volume and surface area are both functions of height, V(h) = pi*r^2*h and S(h) = 2Pi*r(h + r) imply that R(h) = V(h)/S(h) = r*h/[2(r + h)]. r is treated as a constant.