The amount of shampoo you use each time you wash your hair is proportional to the amount of hair you have. Your hair grows linearly over time. Integrate, and therefore, the amount of shampoo you use grows quadratically as a function of time.
One could have a more complicated model of how much hair you have. For example, the probability of a strand of hair breaking at a certain point depends on the age of the hair at that point.
("Shower thoughts.")
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