Old fart here. Didn’t seem that difficult. I haven’t done derivatives in 30 years, and I don’t remember how.
If I were in a hurry, now, I would just brute force it. Of course, if I were fresh out of a calculus class, I would likely have remembered how. Still, brute forcing it shouldn’t give me a good grade, because that’s not the point of the test.
You can use this sort of numerical method to get a good approximation, but it won’t give you the exact solution if that solution happens to be irrational.
I think it was not meant to be taken literally; the editor had been waiting for years to make a pun on “crocodile tears.”
My thinking: good tests include fairly difficult questions to avoid the topping-out-effect, where all reasonably good students get the same score of 100%. I’d guess this question is in there for that purpose.
I think that the fact that the answer is an integer makes this significantly easier than it might otherwise appear, since it makes a semi-naive binary search guess and check process fairly likely to come up with the right answer, since people are naturally going to try whole numbers.
If the answer were, say, 8.013, that would significantly harder to find by the naive approach, and you’d pretty much have to know calculus to solve this in a reasonable amount of time.
I don’t remember any Calculus, but I was able to answer the first two questions with what I remembered of Algebra and Geometry. Can you solve for x without using differentials?
True.
But since in this case the solution is an integer…
Calculus (specifically, derivatives) is what you use to get the equation that needs to be solved for x.
I don’t know of a way to solve the third question using only algebra and geometry (as opposed to calculus or a “brute-force” calculation approach like that described by Bricker or technological assistance).
I’m guessing that may change by country, but I’m used to the default independent variable being x and to both maths and physics using them for a lot many more things than calculating speeds. Your part about being used to derivatives in the form of the speed left me pretty confused for a few seconds.
AKA iterating by bisection, there explained with more formulas.
I agree that using x as the default independent variable is pretty standard in calculus generally. It’s just that in physics, so much early differentiation is with respect to time (velocity = dx/dt, acceleration = dv/dt, the force-momentum relationship F=dp/dt and so on), that I wondered if that was what was confusing them. Just a passing thought, really - it isn’t something I’m particularly wedded to as an idea! 
What might be interesting is to see whether, if this problem was reworded to eliminate this possible confusion, fewer students would be “reduced to tears” and whether more of them would solve it.
That’s my thought as well. It’s not a difficult question at heart, but I feel like they decided to make the question easier by giving the students a step halfway through somebody else’s solution instead of the original question, and the lack of context threw people for a loop.
Seems to me that he zebra Is the one who needs to solve this particular equation.
I’ve forgotten… How does one take the derivative of (36+x^2)^.5?
nm
Use the Chain Rule (or, the Generalized Power Rule, which is what some textbooks call the combination of the Power Rule with the Chain Rule):
The derivative of (something)[sup]power[/sup] is
power*(something)[sup]power–1[/sup]*(derivative of the something)
So it’s .5*(36 + x[sup]2[/sup])[sup]–.5[/sup]*(2x)
= x / sqrt(36 + x[sup]2[/sup])
When I look at the diagram, ISTM that I need to know how wide the river is - the wider it is, the longer it will take the croc to cross over to the zebra side of the river. This should be true regardless of if he crosses in a path perpendicular to the shore, or swims across at an acute angle.
Yet, all the discussion suggests that the width of the river is irrelevant. How can this be?
The width of the river is already baked into the equation they gave the students. Notice that they never directly say how fast the crocodile moves - they only give the equation for how long it takes the crocodile to reach the zebra.
Thanks!
I think there is a way to solve this without calculus, but it involves specialized knowledge that seems to have no connection.
Look at Dudeney’s 536 Puzzles and Curious Problems. PDF http://jnsilva.ludicum.org/HMR13_14/536.pdf.
See problem 311 (pdf page 109, solution pdf page 320).
He basically assumes Snell’s Law to solve it. And it works. (Snell’s Law, which was derived empirically, actually depends on the calculus solution.)
In the Telegraph version, the croc’s path on land is parallel to the shore. This is analogous to light entering a medium of lesser density at the critical angle.
We already have the calculus solution, and the solution this way is more involved, so I’m not doing it.
So if the croc hits the zebra and bounces off, that’s Brewster’s angle? (Assuming the stripes create polarization, of course.) 