Two vertical poles PQ and ST are secured by a rope PRS going from the top of the first pole to a point R on the ground between the poles and then to the top of the second pole as in the figure. Show that the shortest length of such a rope occurs when θ1=θ2.
I'm pretty sure I even did this question, but now looking back on it the answer seems obvious.
Because if you simply take either triangle PQR or RST and reflect it across line QT (the ground), now when you're looking for the shortest length of rope between points P and S, you know from geometry that this is just a straight line. Then you don't even need to do any calculus because θ1 and θ2 are vertical angles so they're congruent.
What bothered me a lot about most of my elementary and middle school math was that you would be taught how to do a certain kind of problem and then expected to follow that method exactly - there was never room for exploration or creativity or anything and an answer was somehow "wrong" if we didn't do it "the book's way." Personally, I think both the calculus and the geometry paths are equally valid and satisfying, just two different ways of telling the same story.
No comments:
Post a Comment