Odds and suds

My life at the moment seems to be revolving around order-of-magnitude or back-of-the-envelope calculations at the moment (to a first approximation, minimizing other effects). You know the sort? The post a few weeks ago comparing thread and shank loading in bolts is really the prime example, as my fascination with metalwork picked up a bit. How do we weld aluminium? What effect do manganese and silicon have on alloys? How much do bicycle axles really flex (the answer: really not very much).

In a similar vein, do you know how often you should cut your hair to break even on shampoo? I spend something like 0.25 pound–hair-inches per three months, with cost per cut of something like £9. Thus:

1/r/.Minimize[{4.5/18(1/(6r)+0.5)+9r,r>0},r][[2]]

produces a happy fifteen months. Any less, and the saving on shampoo is not worth the cost of hairdressing. (I am working on the assumption that shampoo consumption goes up linearly with hair length.)