PoSWW 26: Elevated Concern

Yeah, it’s been a while. We’ve been busy. But, hopefully we’re now back with normal transmission, and we’ll start with an easy one, courtesy of Mysterious Michael. It is an exercise and solution from Cambridge Essentials Year 10/10A.



22 Replies to “PoSWW 26: Elevated Concern”

  1. Maybe this is a *special* 3D space where the triangle inequality doesn’t apply…

    Curved surface perhaps?

  2. Looks like people noticed that 27 is not greater than 28 (if it is equal to then the three points must be colinear) pretty quickly.

    Low-hanging fruit! There’s more to the story :).

    Permission for a followup question? How about if we change the BC side length to 20. Does the presented solution have any issues, or are you happy with it now?

    1. OK, from here it depends whether you are asking about the angle between two planes, a line and a plane or two lines in 3D space.

      I suspect the question is referring to lines, but then without a reference plane, how exactly do we define “elevation”?

      Potentially a nice question for a higher-level class of students, but for now too many invalid assumptions for my liking.

  3. Maybe it is a secret hidden high-level question – determine a surface curvature that makes it true…

    (OK, I’m not sure it is possible to be drunk enough to believe that…)

  4. A minor nitpick: FEBD should be a rectangle in real space, but here the diagram is drawn in perspective instead of with a parallel projection, making FEBD look like a trapezium instead of a parallelogram.

    Another minor nitpick: Why would you ever measure the angle of elevation in a context where you’re using millimetres? The scale of this problem seems inappropriate.

    1. Thanks, edder. Neither had occurred to me, but both are very good points, and I’m not sure they’re that minor.

    1. Ah, interesting. I hadn’t heard of that. I take it no one has any real idea whether or not Heron knew that things had gone wrong?

      1. Heron did not know he had made a mistake. My guess is that the modern notion of a difference a-b was not available then, and that difference meant the amount by which a and b differ, i.e. our|a-b|.

          1. For the very same reasons that the Cambridge authors did not know they made a mistake. If they had known, they would not have included this example. At least Heron would not have done it.

            1. The authors, vetters and whoever wrote the solutions/answers for the question could have easily checked whether or not the triangle was viable in at least three different ways:

              1) Use the triangle inequality. (The side lengths fail).
              2) Calculate the cosine of any angle using the cosine rule. (You get a cosine > 1 or cosine < -1).
              3) Apply Heron's formula. (The product is negative).

              The authors, vetters and whoever wrote the solutions/answers for the question were far too slack. They probably vet VCAA exams as a side hustle.

                1. Sorry, it’s like dynamiting fish in a bucket (but I resist the obvious comment that naturally follows).

  5. 1. If they drew the stuff to scale, they would mess up less like this. (Also, I think it good form to warn the students of non to scale drawings if intentional.)

    2. The curved/warped surface doesn’t work since they said it was in regular 3D space.

    3. It feels a little awkward to tell the kids in problem 13 that the three points are in 3D space, when the 3 points of a triangle define a plane. (If you want E and F out of the plane, just discuss that then and in that problem, 14.)

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