I’m OK with solving for time and distance variables, but I think I’d cry just like the Scottish kids if faced with a problem like this. I get that this is probably some simple calculus equation that math maven could solve in a few seconds, but the way it’s posed makes me want to ask a few questions.
There are no distance measurements given (it seems) for the distance crossing the river before you hit land. How are you to determine what this water crossing distance is which is required to solve the problem?
Can you break that assembled equation down into non-calculus steps so I can see what’s going on?
The equation is time = 5 times the square root of (36+x^2) plus 4 times (20-x), where x is the distance downriver you travel as you cross the river. This equation implies that the width of the river is 6 meters, because the distance traveled through the water is the diagonal of a right triangle with one side x meters long, but you don’t really need to know the width, since you’re given the equation anyway.
If you don’t want to do calculus, just try that equation with different values of x, and plot the results, and it will be pretty clear that the minimum value for time is when x=8 (and if you check your result by trying 8.00001 and 7.99999, you’ll be pretty sure that varying from 8 always makes the travel time longer.
P.S. I did do the problem the calculus way, and would have gotten it right except for a silly algebra mistake…
If they were allowed to use graphing calculators on the test, all you need to do is plug the function in and look at the graph, which looks like this. Just scroll along it and see that the min point occurs at x = 8.
The “minimum” crossing distance is only theoretical, and is not needed. For part a.ii you are just plugging zero into the function to get the time for “swims the shortest distance possible”.
You don’t know the distance, but you also don’t know how fast the crocodile swims. What you can figure out is how much time it would take to swim straight across the river. This is enough, since all you are concerned about is the shortest time. The larger the distance, the faster the crocodile must be able to swim.
BTW, that time is 3 seconds: 5 * sqrt(36 + 0) = 30 tenths of a second.
As for the whole problem, I would not have remembered enough calculus to do it without graphing. And I find that Google’s graph is weird: it uses discrete values for f(x) instead of for x. This made it much harder to see if x=8 exactly was going to be the answer.
This is so true. It doesn’t say what age these students are, nor what calculators they are allowed to use. With the graphic calculators students often use nowadays, it is a piece of cake (or should be). Even using calculus it is very doable, and definitely something I had to be able to in high school.
Huh? f(x) is a standard way of representing a function. In the original problem they call it T(x) for time, but it’s the same thing. The graph clearly shows x = 8 is a minimum with f(x) (or T) being 98 = 9.8 seconds. No idea how that managed to confuse you.
I always used a graphing calculator on all my tests in high school, and they’ve been using them for many many years since before I went to school. I would be pretty surprised if the students didn’t have them.
I think the children would be 16 or 17 and certainly it wouldn’t have been set on the paper had it not been taught in the class.
I don’t know anything about calculus so would have been stumped but the explanatory video broke the problem down into steps that were easily understandable even to me and had discrete solvable parts. I’m assuming that such an approach (and the algebra to go along with it) would have been the way it was originally taught so it shouldn’t have been too much of a trial…except…
people often get into a tizzy when the maths is put into a real world situation and you have to decode the question in order to be able to perform the correct functions in the correct order. In this case the skill is not in plugging the figures in to the equation but in thinking clearly enough to know where the figures should come from.
What struck me about the problem were
[ul][li] The most interesting and important part of such a problem is developing an equation from a word problem. Can the student apply the Pythagorean theorem? Does he know what to do with velocities? And so on. Instead a final equation was just presented. In fact the relevant words (width of the stream, velocities) were not present at all, just implicit in the equation.[/li][li] Part (a) should be easy for high school freshman; part (b) difficult for high school seniors. I guess the idea is to let mediocre students accumulate a few points, but why?[/li][li] Is plotting the graph on a calculater and looking for the minimum really the way things are done today? Strikes me as wrong, but I’m old-fashioned.[/li][li] I think there is an elegant path to solution using neither a graph nor explicit calculus. I doubt the boring Youtube presents it.[/li][li] The problem is almost identical to the very important Snell’s Law named 400 years ago after Snellius (but according to Wikipedia actually discovered by a Baghdad mathematician in 984 AD. The problem could be presented in a way to have much pedagogic value.[/li][/ul]
Strictly speaking, iterative methods (“trial and error”, if you must) won’t establish a definite minimum - it will only show that 8 is a better approximation to the solution than, say, 8.1 or 8.01 or whichever alternative value you actually try. But the function’s not hard to differentiate if you know the chain rule (and impossible without it).
You can easily solve it for cases where x = 0 (which they ask for) and x = 20 (which they don’t), and so know the range. It’s some relatively simply differential calculus to get the equation for the minimum, and the constants are chosen so that all the square roots come out as integrals. If you knew or suspected this, it might cut down on the number of cases you’d try if you were doing it by trial-and-error.
I don’t know exactly what level of difficulty they were aiming for, but this one is far below the level of difficulty of the problems we worked in our Math League competitions in high school. And we had to solve ten problems in an hour.
I didn’t want to bother with the calculus, so I used iterative approximation. To save time, however, I started with the correct answer, because social engineering is a skill, too.
The first two questions ask for T(0) and T(20) which are trivial. The last part is also easy with calculus, but much harder without. The equations imply that the croc moves 50 and 40 meters/sec in water and on land. There is a way of doing that imagines that the land distance is expanded by 5/4 and the speed is the same but it is pretty subtle. The same idea can be used to explain refraction of light as a result of change in the speed in different media. But that kind of reasoning is not to be found in an exam unless it has previously been explained. But without knowing what the students were expected to know it is impossible to say if it was reasonable. That illustrates the power of calculus. Subtle becomes trivial.
