Disparate puzzles with same solutions

[bonanova]

I notice that two disparate problems can be solved by the same equations. The chestnut about adding 1 foot to the length of a tight band around the equator and raising the band with a pole as high as possible at one point is one; the optimal escape path for the girl rowing a boat in a circular lake (recently posted here) is another.

Eventually you get the equation

Tan(x) = x + c

where c is some value you can calculate. You need x because the answer to the problem is

Cos(x) = answer.

I wonder if there are other problems like this, or if there is a prototypical problem that this analysis describes that can be flavored to generate a family of seemingly different problems. Or does that question even make sense?
_________________
_{Vidi, vici, veni.}

[L'lanmal]

If you are looking for a family of puzzles which all have the same equation as an answer (and not just the particular equation you described), you might want to check out exercise 6.19 of Stanley's Enumerative Combinatorics. The exercise is to prove that 66 different sets are all counted by the same formula (namely: the Catalan numbers).

[Nsof]

Fibonacci numbers creep up in many puzzles/problems.
_________________
Will sell this place for beer