My nephew showed me an exercise from his school-textbook about boiling eggs. Here is the exercise (translated from german):
- To make one hard-boiled egg in a pot of water one has to put it for 8 minutes into boiling water. How long does it if you put 10 eggs into the boiling water instead of one to make hard-boiled eggs?
- Using an electric egg cooker (something like this: https://www.buyegggenie.com/) one needs 7 minutes to make 3 hard-boiled eggs. How long does it take to make 6 or 10 hard-boiled eggs?
The chapter in the textbook is about simple calculations with proportionalities ("Dreisatz" in german). For my nephew it was not problem to solve the exercises if one assumes that the cooking time and the number of eggs are proportional. But for him and also for me it is not clear why and when one can assume proportionalty and how to justify it physically.
My intuition (which is probably wrong in this case) says that in both cases we don't have a proportionality moreover that the cooking time is independent of the number of eggs.
To the second part of the problem: I don't own a electic egg cooker. From googling around I have found that it works with water steam (whereas in the first case the eggs are in the water). And that one needs less water in the electric egg cooker to boil more eggs. It is not clear to me how to interprete the second problem. One could assume that one uses always the same amount of water in the electric egg cooker or that one fit's the amount of water to the number of eggs (which will result in different cooking time – number of eggs relationships, perhaps one proportional and one constant…)
So it would be great if someone could clarify this physically. I.e. how does the cooking time depends on the number of eggs and why and under which assumptions in each case.
How does the situation change if one puts the eggs not in the already cooking water but into cold water at the beginning.
Are there methods of making hard-boiled eggs such that the cooking time is proportional to the number of eggs and are there methods such that the cooking time is (roughly) independent of the number of eggs? How to justify it physically?
Best Answer
In the real world, you are approximately right and the cooking time is pretty much independent of the number of eggs. An egg only cares about the temperature of the water that surrounds it and as long as it is kept near the boiling point at all times, an egg can't possibly "know" how many siblings it has. So its gradual change (and its rate) is universal, independent of the number of eggs.
On the other hand, if this is some textbook example about proportionality, it's one of the would-be real-world applications of proportionality and the underlying assumption is that the cooking time is proportional to the number of eggs. In the real world, this can only be true if the number of eggs affects the actual temperature of the water around them.
A justification of the proportionality is that the egg cooker needs to pump a certain amount of energy $E$ to an egg to turn it into a hard-boiled egg. Because the egg cooker consumes the same amount of energy per unit time, it needs $N$ times longer time to pump the energy $NE$ into the $E$ eggs. In practice, this may be a somewhat reasonable description if the egg cooker is weak enough and it has a hard time to achieve or maintain the boiling point as the eggs are cooling the water down.
The truth in the real world will be somewhere in between: the cooking time may grow with the number of eggs but the growth is much slower than the direct proportionality.
When I am cooking various deeply frozen lunches, the recommended cooking times for half a bag are often something like 80% of the cooking times for the whole bag. The idea is that with more food on the pan, it takes a bit longer for the heat to penetrate from the pan to the interior of the food. The idea with the eggs is similar if the cooking time is supposed to increase with the number of eggs.