optimize travel time on a 2d plane

Question

There is a dog at (x,y) = (37,10) with units say feet. The dog wants to go home located at origin (0,0). The dog speed is 6 feet/seconds on the x,y axes and 6.5 anywhere else. What's the fastest path to the origin?

My answer says a direct path but that seems to easy (this is supposed to be a trick question). Am i missing something?

I don't think you're missing anything -- the speed on the $x$ and $y$ axes is slower than the speed everywhere else, and the straight-line path avoids the axes anyway! — diracdeltafunk, Mar 01 '24 at 21:43
Probably the trickiness is that the question tries to suggest that something tricky has to be done, while that isn't the case... — Jos Bergervoet, Mar 01 '24 at 21:44
Are you sure you don't have the speeds backward? If not, then a straight line is the answer. Your teacher may want you to prove that by considering paths to the axes and minimizing the time as in the derivation of Snell's law. — John Douma, Mar 01 '24 at 22:11
For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. — Henrik supports the community, Mar 01 '24 at 23:40

score 0 · Accepted Answer · answered Mar 01 '24 at 22:07

Optimization should be with a constraint, here no constraint. Tag is wrong.

Fastest path length is the straight line length $\sqrt{37^2+10^2} $ whatever its speed while proceeding to the origin directly or along axes, no trick. Divide by speed to get the time of dog's walk if required.

optimize travel time on a 2d plane

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