WitCH 57: Tunnel Vision

The following is just a dumb exercise, and so is probably more of a PoSWW. It seems so lemmingly stupid, however, that it comes around full cycle to be a WitCH. It is an exercise from Maths Quest Mathematical Methods 11. The exercise appears in a pre-calculus, CAS-permitted chapter, Cubic Polynomials. The suggested answers are (a) \boldsymbol{y = -\frac{1}{32}x^2(x-6)} , and (b) 81/32 km.

7 Replies to “WitCH 57: Tunnel Vision”

  1. I’ll start with the easy targets:

    x and y are never defined, let alone properly defined.

    4km from the origin? I assume they mean a horizontal distance? Similarly, “above this level” – which LEVEL exactly?

    “Greatest breadth” – why not just say “width”?

    Bounding curve? Yeah… um…

    1. Thanks, RF. Yep, the x and y are irritating, particularly the x near the intercept, where supposedly x is. And yes, “breadth” is a hell of a word; I’d have somehow tried to go for “length”, but even now I can’t get a clear picture in my head fo what was intended. And, yes, “bounding curve” is just hilarious.

  2. Another contrived ‘real-life context’ bullshit question.

    We’re meant to assume the “level” in part (a) is the x-axis, I suppose. But the x-axis, relative to which these ‘measurement’ are made, is not defined. No worries.

    Then we’re meant to assume that the breadth is measured along the x-axis and assume that the point (6, 0) lies on the curve. No worries. It looks like (6, 0) and we all know pictures are trustworthy. (Even VCAA wouldn’t be this amateurish).

    I won’t even comment on what a stupid cross-section the tunnel has, although I’m sure we could imagine adding all sorts of geological bullshit to make it seem reasonable. (But a tunnel 1 km high at its greatest height, that’s one fucking high tunnel!! About as high as the pile of bullshit in this question). So no worries.

    After making all these \displaystyle assumptions I suppose we can finally define \displaystyle f(x) = ax^2(x - b) on our CAS.

    Then for part (a) we can ask our CAS to solve f(6) = 0 and f(4) = 1. We get the DQ* answers for a and b so all is well. It’s a pity that the most students will use a CAS rather than solve for a and b using simple algebra. (In fact, I’d argue that doing it ‘by hand’ is faster than using a CAS).

    But for me the real trouble is part (b). I \displaystyle assume we want f(7) where f(x) = -x^2(x – 6)/32 (that is, the equation of the curve is the same as that found in part(a)):

    f(7) = -49/32 therefore the greatest height is 1 + 49/32 = 81/32. That’s a relief, we’ve done it again. But holy shit … that height is about 1/3 of the cruising altitude for most commercial planes. So much for reality.

    Well done, DQ*. Plenty of assumptions have to be made and we have a contrived ‘real life’ context that is nothing like real life. Another excellent example of complete bullshit.

    * I have my own alternative name for this particular textbook, which would probably expose Marty to litigation, so I will simply refer to it as DQ. But it does have the redeeming feature of being really colourful.

    1. Thanks, JF. Part (a) is bad enough: I’m still not sure I understand what “cross section of the plan” means, if anything. For ages I thought the curve gave the cross section of the mouth of the tunnel, but of course that is dumb. But everything is dumb. What’s the tunnel doing at x = 6? Smashing into the ground? Note that (a) simply asks you to assume that the maximum is at x = 4, because why not, so no CAS is required.

      And yes, (b) is simply incredible, complete with shrunken drawing for the expanded curve. It took me forever to figure out they want you to evaluate the same function at x = 7, and compare to the x = 4, because what the hell does that mean? It’s completely insane.

  3. Just a follow-up on JF’s general remarks on the Maths Quest Methods 11 textbook, and the texts I choose to whack. Yes, Maths Quest 11 is a bad book, and in certain systemic ways it is appalling. MQM11 probably deserves whatever nickname JF has for it (although I’m glad JF is keeping it to himself). I will post another couple WitCHes from MQM11 in the next day or so.

    On the other hand, and to my surprise, there are a few ways in which I prefer MQ11 to Cambridge Methods 11. To begin, MQ doesn’t have the snotty, more-mathematical-than-thou tone of CM11. True, part of the non-snottiness is just from a general sloppiness, and more precise language was definitely required. But on other occasions MQM11 seemed genuinely to be pushing properly intuitive reasoning, where CM11 nearly always errs on the side of pointless rigidity. Another aspect of MQM11 I generally liked were the short historical introductions to each chapter. Their connection to the chapter content is typically tenuous, and the grasping-at-straws attempts to include female mathematicians is embarrassing; but the histories in and of themselves tend to be well-written and interesting.

    As for which textbooks I choose to whack: I whack whatever I have access to whack. As it happens I have access to the Cambridge texts, and so I whack them most often. That is fortuitous, since Cambridge is the most respected and most mathematically solid of the Victorian texts; demonstrating the 2nd ratedness of these texts is a default upper bound for other texts as well. But I’ll whack anything and, as it happens, a copy of MQ11 recently appeared on my desk.

    So, if anyone wants me to whack their text, get a copy to me and I’ll see what there is to whack (and, less likely, to praise).

    A final couple of notes on these textbook WitCHes. First, I am trying to avoid cheap shots; generally I try to choose the WitCHes that indicate the worst aspects of the textbook as a whole. So, they are stark examples, but not unrepresentative examples. (This MQM11 example here is a bit of an exception and, as I wrote, is really more of a PoSWW.) However, and secondly, the focussed nature of a WitCH is both a plus and a minus. The plus is that a WitCH encourages the reader to look with proper care at the presentation of a very specific topic; what is really said, and whether what is said is confusing or misleading, or straight-out false.

    The minus is that a WitCH cannot give a proper sense of the textbook as a whole. What is required for that is a proper, and properly long, review. I’ve done a couple of those, and am not 100% against doing more. But I’m about 98% against doing more, for one simple reason; the publishers should be arranging this, and don’t, and the idea of doing their work for free is really fucking irritating.

    If a publisher hired a competent and attentive mathematician to carefully review their text, and if the mathematician’s review was respected, there is simply no way a text like MQM11 or CM11 would see the light of day. No competent mathematician paid to do a detailed, competent job, would sign off on these texts.

    1. “If a publisher hired a competent and attentive mathematician to carefully review their text, and if the mathematician’s review was respected, there is simply no way a text like MQM11 or CM11 would see the light of day. No competent mathematician paid to do a detailed, competent job, would sign off on these texts.”

      The review might achieve in getting some small changes made, but I suspect the response would be
      “Thanks for your time, the cheque’s in the mail, we’ll pass this on to the writers”
      and nothing of great significance would actually get changed.

      However, if it became known that you vetted the textbook, it would be your reputation that might suffer in such a case (particularly if the textbook added your name).

      1. Thanks, JF. I consciously included “if the mathematician’s review was respected”.

        I don’t know if the “you” refers to “Marty” or “hypothetical mathematician reviewer”. In any case, I think you have a very good idea there: a mathematics textbook shouldn’t be published without a reviewing mathematician willing to have their name published alongside. Of course not all reviewing mathematicians will be competent, but it’d be a good start.

        As it happens I’ve been approached by exactly two authors and one publisher about reviewing or consulting or whatnot. Neither of the authors replied to my (free) critique; I talked about one of these episodes in my Hell presentation, and I plan to write about it on the blog. The meeting with the publisher was different and very good; he clearly wanted to involve mathematicians, and I understand he has engaged a good person in this capacity. I don’t know if it will end up making a difference, which is one reason why I declined to be involved, but there appeared to be some decent level of sincerity.

        I don’t think it is surprising that I haven’t been approached more. The maths ed world tends to regard me as a lunatic, and they’re either contemptuous of me or scared of me. But there are other mathematicians around, and the question is why almost no publishers approach these less ogreish mathematicians. And the answer is trivial: the publishers don’t give a shit. They simply don’t care if their textbooks are bad.

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