Assume that a cup of coffee is a cylinder. The coffee machine at my workplace always produces the same amount of coffee, so the volume is constant. The coffee is always really hot, so I'm looking (out of curiosity) to maximize the cooling effect of the cup. If the coffee has a bigger contact area, it will transfer more heat per unit of time. Assume that the amount of heat per time transferred with cup and air are the same, or consider a "closed cup", for example. Assume it is not relevant.
I have some cups to choose from. What is cup that provides the biggest contact area (with air and cup), given that the volume is the same?
Theoretically I know that we could either have:
- a 1 atom height cylinder with a large radius or
- a half atom radius with a large height.
However these are some extreme and particular cases. I'm looking for a general formal proof that could give me some rationale for my choice.
Something like "you should choose whatever cup has more radius/height",etc.
Sorry for any possible lack of formality. This is my first question here. :)
The "cooling area" is given by $2\pi r^2 + 2\pi rh = 2\pi r(r+h)$
So the optimal choice is the cup with the biggest value of $r(r+h)$ and fixed volume $V = \pi r^2 h \Rightarrow h = V/\pi \cdot r^{-2}$, $$r^{\text{opt}} = \mathop{\rm arg\,max}_{r>0} r^2 + V/\pi \cdot \frac1r = \mathop{\rm arg\,max}_{r>0} r^3 = +\infty$$ Answer: Take the biggest radius.