I would've arrived at the teacher's solution, but the question allows different interpretations and both answers are correct assuming different interpretations.
The correct answer would be "I do not know, this problem is under-specified."
The correct answer would be "I do not know, this problem is under-specified."
There is no sufficiently logical way to get to any particular number other than 20; the shape of the plank does not allow you to cut across and make your cuts intersect like you might with a square board. There is no reason on this particular shape to prefer "15 minutes" over "14 minutes" or "25 minutes". It all gets lumped into "any amount of time whatsoever".
If "any amount of time whatsoever" was an acceptable answer it wouldn't make sense that a single cut takes 10 minutes, so we should discard that answer. This leaves only one candidate answer, 20 minutes.
*"any" would be limited by how long of a diagonal you can make but it would be hours
I obviously thought 15 min when I first read it and my brain didn't want to accept any other solution until I read the post below where it said 20 min and explained it as 2 pieces = 1 cut = 10 min, 3 pieces = 2 cuts = 20 min.
And now I can't see why my first thought was correct. Did you come up with some good rationale as to why it should be 15 min or other?
Because it depends on whether you 1) require that the N pieces be congruent and 2) what counts as a cut. I think the textbook answer is based on assuming 1) no, and 2) cutting along a line segment at least as long as a side.
Alternately, what counts as a "board" and a "cut".
Then you get the answer by assuming you cut a square board in half, then one of the pieces into squares (which requires cutting along a line segment half as long).
The problem does specify 'works as fast' without any regard for length, though. And it obviously isn't actually a geometry problem because it doesn't even specify any ratios or angles - you could just cut a corner off and be done in a few seconds!
If it was dressed, that 1 might be very small indeed.
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There is only one logical interpretation (though you'd have to actually think past the conclusion your brain jumps to), the question was perfectly clear about the board being cut into two pieces.
In that case, I'd write my assumptions about the problem and show how I arrived at the solution. If the teacher still says it's wrong, I probably won't bother arguing - I don't waste my time arguing with morons.