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.

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# PoSWW 26: Elevated Concern

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22 Replies to “PoSWW 26: Elevated Concern”

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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.

Looks like the writers just CBF checking to see if the distances were possible.

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

Curved surface perhaps?

I’d say warped rather than curved.

To paraphrase The Matrix: perhaps there is no surface.

Aha! So we now have a list of suspects who might have written the 2019 Specialist Maths Exam 2 MCQ12.

A reference for such writers:

https://en.wikipedia.org/wiki/Triangle_inequality

Looks like people noticed that is not greater than (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 . Does the presented solution have any issues, or are you happy with it now?

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.

Ah, I think I get your point. And a very good point. But I don’t see a problem with 20.

Neither do I now that I actually crunched the numbers.

Rookie error.

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…)

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.

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

If you google for heron and complex numbers you will see that this has happened to greater men

(I am talking about the nonexisting frustum in his stereometria).

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?

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|.

How do you know Heron didn’t know?

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.

Yeah, probably right. Still, it was a long, long time ago.

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.

Ugh. You don’t have to worry the VCAA bone every time.

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

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.)