A crocodile, a zebra and an equation. Can you find the answer?

October 9, 2015 • 11:58 am

by Matthew Cobb

A recent Scottish Higher maths exam question – designed for pupils at the end of their schooling – proved a bit too hard for the students, and the pass mark for the paper had to be dropped to 34%. Can our readers do it? I could just about work out the first two answers, but the main part was way beyond me.

Here’s the paper. To help out the maths challenged, I’ve posted a 10 minute YouTube video from DLBMaths in which he goes through the question and the answer, in a lovely Scottish brogue (the first minute is some technical stuff about the format of the exam – most of the video is about the problem).

And here’s the answer. Don’t watch the video until you’ve had a proper go at the question! As always with maths, it looks so easy when it’s explained carefully.

 

63 thoughts on “A crocodile, a zebra and an equation. Can you find the answer?”

  2. Some reasonable inferences.

    (20-x) is the land distance so 4(20-x) measures land time. Note this is an inference.
    5(square_root) then measures the water time. The square root will be a dstance, the path across the river shown, which is the hypotenuse of the triangle with side x and width of the river. From pythagoras the river is 6 feet wide. If he swims the whole way then we should use 5*square_root(36 + 20*20)

    The minimum can be found by taking the derivative wrt x and setting it to 0.

    1. I’ve just noticed something that is now annoying me.

      The time taken to cross the river is a constant times the distance swum. We can calculate from the fact that the constant is 5 that the crocodile can swim at 2m/s.

      The problem is that the zebra is 20 metres upstream which means there must be a current. The crocodile’s speed will depend in part on its angle relative to the current and that is not in the equation.

  4. Ha! Even 55 years after taking differential calculus, I remembered how to do this. Amazing. But those kids are starting early.

    1. Are they really? This is for 17-yr-olds who are studying maths. And it’s a standard calculus technique, drummed into anyone studying calculus. It seems to be about the expected level to me (I think the maths-able kids started this stuff about age 15 in my cohort.)

      1. Back in the Pleistocene, I had this kind of thing (differential and integral calculus) at age 17 in US high school.

        It’s not a very hard word problem. And finding the diff or where it’s = 0 isn’t very hard.

        What would have tripped me up (and did) was that the time is in 1/10 sec instead of sec. But looking that the measurements (20 m) should have cleared that up.

        I’m just grateful there wasn’t a factor of 2 in there somewhere! 🙂

  5. a is easy. b is tough, but possible for people at that level. I’d expect only the top students to get it, but it’s fair.

    It’s hard to remember whether I would have been able to solve it 7 years ago when I took my A Levels, but I’d like to think so. I’ve barely done any Calculus for 3 years and didn’t remember things that I used to know off the top of my head but I still got it fairly easily. Theoretically it should have been easier when I was doing this sort of thing every day.

    1. Of course that raises a good point. If all the questions are easy enough for everybody taking a maths paper, then everybody will get 100%.

  6. The ‘x’ designation in the equation replaced by ‘P’ would have made the problem clearer for me.

      1. …and is it the favored prey of the spiders or the venomous mammals? I mean…Australia! Of course the crocodiles are the least of your worries….

        b&

  8. My calculus is fresh enough to know that you have to take the derivative of the function and find the point where it equals 0.

    But my calculus is rusty enough to have to look up how to do the derivative of sqrt(36+x^2)

    However, the famous online graphing calculator
    desmos.com/calculator allows one to easily graph y=5sqrt(36+x^2)+ 4(20-x)
    and one can simply eyeball that the minimum of the function occurs at x=8

    1. Thanks for he tip about the online graphing calculator. My calculus is so long ago and so unused that my first instinct was to just pull out some graph paper and start plugging in some values between 0 and 20 until I got enough sense of the shape to guess the right answer.

    2. Totally with you on how rusty my calculus was! (Excel and FEA have done away with the need to actually use it anymore.)

      I threw it into excel and had the answer and the graph in a few seconds.

      1. I think relying on plug and chug shows a grasp of math. I knew the answer was between five and eleven just with a little doodling.

        And all of this ignores that some people might have real experience with crocs and gators. The gators are restless where I live because a wet spring and early summer have been followed a dry late summer and fall. Gators gotta get a few good meals soon to over winter. In real life a gator will always choose to attack from the water. The only gators that attack on land are females near their nests.

  9. That’s snells law. Which in optics describes de path followed by the light when it crosses several different mediums (vacuum, water, lens, etc) and explains refraction and so on.
    The light does basically the same as the crocodile. It “chooses” the quickest path between “A” and “B”, which as the problem shows is not necessarily the shortest.
    A tough problem for a “higher”, tough. I was asked the very same question in an exam in my old days back at Univ. (1st course)

    https://en.m.wikipedia.org/wiki/Snell%27s_law

  11. got it in about 3 minutes. glad to know the math noggin still functions. Not sure if the description of “higher maths” & “end of schooling” means… but it seems to be a pretty easy problem, as long as the high schoolers are acquainted with pretty basic differential calculus. (i.e. no analytical geometry or logarithms / transcendental functions involved). At the end of my high-schooling, I chose “math analysis” over calculus, so barely got this far. I probably wouldn’t have been able to figure this one out until I was done with my first semester of college, for that reason.

    1. In my high school “math analysis” was required prior to taking calculus. I found it more difficult than calculus. Although early calculus was pretty tedious, what with learning all those proofs. I found more advanced calculus much easier.

  12. BTW, I once threw this problem at a few colleagues, with some numbers thrown in for A and B, and asked for h to something like 16 significant figures.

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

    my answer involved Newton’s method and a computer running GBasic & double-precision. It starts off easy, then gets messy really quickly. Illustrates why the maths behind the electron/proton attraction/repulsion energies in a simple hydrogen molecule devolve into approximations. Similar problem.

  13. First two parts are easy (x = 20 and x = 0).

    The third part should be easy for anyone taking that test. The function is not linear (the variable has an exponent), so the minimum value is either where ‘x’ is at the min or max of its range (0 to 20), or somewhere in between where the slope of the function’s graph changes from negative to positive. So take the function’s derivative and set it equal to zero, then solve for ‘x’. As it happens, I have long since forgotten an easy way to take the derivative of that function, but that shouldn’t be a problem for the students in question.

    1. The third part is hard, particularly in an exam situation. It’s not the easiest function to differentiate.

      That said, it’s fair, and the smarter kids should be able to solve it.

      1. You have to understand that when you are 17/18 and doing this exam, differentiating functions like this should be meat and drink. When I was doing maths at that age, we practised differentiating things like this over and over until it came naturally.

        This is differentiating a function of a function. The student will immediately think “chain rule”, then it comes down to differentiating a square root and a quadratic.

        What may have made the problem confusing was the application to a sort of real world situation. When I was learning this stuff, it was all done in the abstract. Sometimes people have trouble mapping mathematical techniques to real world situations.

        1. Your last paragraph is very relevant. As an engineering student I found exactly what you said – it was sometimes quite surprisingly hard to match the idealised formulae to actual (sometimes messy) real life. Amongst other things, it requires a judgement of which factors are critical and which can be ignored. In the crocodile example, for instance, is the time it takes for the croc to climb out onto the bank sufficiently small that it can be ignored? (Obviously, from the way the question is posed, the official answer is ‘yes’ – but that might not have been apparent if the problem had been worded differently).

          cr

        2. I think you’re overestimating the mathematical ability of your average 17 year old. I found the question easy, but I have a Physics degree. I’d have found the question easy enough back when I was taking A-Levels too, but I took A-Level Further Maths, which had many questions far harder than this one.

          For the people taking this exam, I’d bet that at most only the top 10% of students got it right, probably less. This question is being used in the media as an example of how the paper was too hard. That is clearly ridiculous, as it should be easy for the top students, but just because you found it easy, and I found it easy, and the kids going on to do Maths, Physics and Engineering degrees probably found it easy, doesn’t mean that most kids that age would find it easy.

          1. But isn’t that the point of exams? to differentiate (sorry) between the students. So you have to construct questions that most students will be able to so some parts of, but also most students will at some point start floundering and losing marks.

    3. The chain rule is what you need for the derivative. Any student who just took calculus should be able to spit out the chain rule or they deserve to fail. I don’t know if the Scottish students in question studied calculus, but if they did, then lowering the standard is a shame. If they didn’t, well, kudos to those students who figured it out without knowledge of calculus.

      P.S. I worked it in a couple of minutes myself. Although not hard for me (I am in a mathy field) I still enjoyed having a random problem presented to me. Makes me think I should start doing more problems for fun… maybe this could be a regular feature?

  14. There is another pet peeve. ‘Maths’ or just ‘math’? Which is preferred?

    3. Yeah, well, I’m an engineer too but that’s cheating. 😉

      Three graduate students were given the problem of finding the volume of a red rubber ball.
      The mathematician measured the diameter and evaluated a triple integral.
      The physics student submerged it in water and measured the volume of water displaced.
      The engineer looked up the answer in his red-rubber-ball table.

      cr

  16. This question isn’t very interesting. It’s tests a student’s ability to plug a number into a formula to answer the first part.

    The second part checks the student’s ability to take a derivative, set the derivative to 0 and then solve for X and plug the result back into the equation. If you have been studying the subject matter very little thinking is necessary to solve this.

    The problem is very disappointing because they could have made it more interesting by giving the width of the river, the velocity in water and on land and then had the student come up with a formula for the total time and then solve the two parts of the problem.

    They have the croc moving faster on land at 2.5 meters per second compared to the water at 2 meters per second. I would think crocs are faster in the water but who knows. I’ll have to check that out.

    2. Yes, I agree with you. The most important thing nowadays is to learn people how to write the equation(s) that models adequately the problem(s). Derivation is no problem for the functions of A or I calculus, mainly because programs like mathematica, maple and others algebraic manipulators are available and free online (http://www.wolframalpha.com , for example).
      Having said that, at least one can make the problem above more interesting adding two other things:
      y axis
      |———***(croc arrives at zebra)
      |———+++Z
      |—-++*++—-
      |++++**——-
      |–**———
      |**___________> x axis
      |croc at (0,0)
      – One margin lies at y=0 and the other diverges approximately like a straight line : y=a*x+b (++ in the graph)
      -The zebra lies (Z) at margin y_zebra=a*x_zebra+b.
      If the croc starts at (0,0) and swims in a straight line y=K*x (the ** in the graph), what would the the value of K that minimizes the time? (Croc’s velocity in water (Vw) and land (Vl) are given).
      If we consider that the water can have its own velocity downstream, what changes in the modeling equation?
      Math can be fun…

      1. great comment!
        There are many (MANY) tech tools that can solve plug&chug-able equations. The MOST important things is that students (people) can ask the pertinent questions and set up the mathematical model…how fast on land? on water? do I need a triangle? does river speed matter? etc.

  17. Scottish Highers are taken at the end of the fifth year of High School. So students would be 16 or 17.
    It’s a long time since I took Higher Maths (1980) but this seems to be much the same level of difficulty as then.

    1. But it has a crocodile in it… as you do the derivative you can just *feel* it bearing down on you, ready to rip your leg off with it’s razor sharp teeth…

    2. You obviously never met the realisation that when deltax approached zero, deltay/deltax actually had a value.

      I remember being gobsmacked by this.

  19. Teaching 1st year physics undergraduates at university, the VERY FIRST lesson was how this question is nonsense, dimensionally.

    T = sqrt(w^2 + x^2) / v1

    where w=6m, v1=Q/5, etc.

    1. I don’t see a problem. The equation is correct. Distance divided by velocity equals time.

  21. My right-brained mind doesn’t do well with math and I never enjoyed the subject much. I wasn’t terrible (no C’s, mostly B’s and a few A’s) but when I met my high school core requirements (advanced algebra), I didn’t go for trigonometry or calculus. Since my major was English Lit., I escaped math in higher education.

  24. The answer should be irrelevant because an adult zebra can definitely outrun a crocodile!

    The most difficult part of this question would be coming up with the equation, as a mathematician I am disappointed that they have taken out the most difficult part. It is pretty easy once they give you the equation (for a math person). It could have been phrased entirely as a word problem involving the speed of the gator on land, water, and the speed of the current, if any.

  25. Well (a) is easy, just substitute X=20 and X=0 in the expression.

    (b) requires taking the derivative of the expression, I think, to find the minimum, and my calculus is 40-years-rusty (the sqrt(36 + x^2) is what floored me).
    Unless there’s some sneaky trick built in to the way the question is posed (like the well-known one about ‘a train leaves Cincinatti for St Louis 400 miles away at 40 mph. An hour later a train leaves St Louis for Cincinatti at 60 mph. How far from Cincinatti do they meet?’ – where the trick is NOT to calculate distance covered, but time)

    cr

  26. i) x=20 m, so T=5*SQRT(436) 10ths or about 10.4 s
    ii) x=0, so T=30+80=110 10ths or or 11 s
    iii) Set dT/dx=0; after starting at it for a while, x=8 m; T=9.8 s

    Not much difference, so the zebra is screwed no matter what path the croc takes.

  28. Crocodiles surely lie in wait in the water – there is no way it is going to chase after a zebra on land. Therefore it should swim submerged all the way.

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