A flea's shortest escape across a box

HardBrainteasers Calculus~9m

A flea sits at one corner of a solid box with side lengths $2$ , $3$ , and $4$ , and must reach the diametrically opposite corner, crawling only along the surface. It moves at $2.5$ units per second. What is the minimum time (seconds) to reach the opposite corner?

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Unfold the box flat; the path becomes a straight line of length $\sqrt{(a+b)^2+c^2}$ .
Minimize over pairings: $\sqrt{(2+3)^2+4^2}=\sqrt{41}$ ; divide by $2.5$ .

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2.5612 (± 0.01)

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A flea's shortest escape across a box

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