The basic idea behind finding the solution is simple: identify relevant variables, apply any constraints to find any relations between the variables, check boundary/edge cases then use calculus to find the value of the variable/s that minimise the length then use the value to find the minimum length.
Nothing exceptional here, in principle, though there is plenty of room for error in the fairly large amount of algebraic manipulation involved, until we get to the point of having to find real solutions to a quartic. Here we have to be able to spot that this is a special case of a quartic that can be solved without recourse to the quartic "formula".
Learning to spot such "nice" cases is one of the objectives of the training that MO competitors undergo.
(I should point out that I have not had such training, and the idea of completing the fourth power came to me after several days of contemplation, in my sleep)
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