An Optimization Problem
I don't get a haircut that often. When I do, I want it buzzed so that I'm practically bald because I want to get my money's worth. A while ago, I thought I might be able to save money if I simply did away with haircuts altogether. At $15 a pop, this could amount conservatively to $3000 over the course of a lifetime or $260,000 adjusting for inflation and compound interest. I could literally buy a house in cash. My hair grew pretty long until I realized I was using a lot of money on shampoo. In a few decades I'd be going through a bottle every week just to keep my hair clean.

This raised the question: to save cost what is the optimum length of time one should wait between haicuts? I'll spare you the calculus. Suffice it to say the function for total cost graphed like an ascending sawtooth of sorts. I took the slope at the point immediately after the first haircut to smooth things out for differentiation. Consequently, the formula could be off given short periods of time. But over a lifetime, it approaches correctness. The formula is as follows:

Optimum length of time between haircuts = ((2*H)/(S*E*R))^.5


Where
S=cost of shampoo ($/weight)
H=Cost of haircut ($)
R=growth rate of hair (length/time)
E=amount of shampoo it takes to wash hairs (weight/length)
At this point in the problem, I realized I'd actually have to do some measurement and gave up.