This one’s shooting a smelly fish in a barrel, almost a POSWW. Sometimes, however, it’s easier for a tired blogger to let the readers do the shooting. (For those interested in more substantial fish, WitCH 2, WitCH 3 and Tweel’s Mathematical Puzzle still require attention.)
Our latest WitCH comes courtesy of two nameless (but maybe not unknown) Western troublemakers. Earlier this year we got stuck into Western Australia’s 2017 Mathematics Applications exam. This year, it’s the SCSA‘s Mathematical Methods exam (not online. Update: now online here and here.) that wins the idiocy prize. The whole exam is predictably awful, but Question 15 is the real winner:
The population of mosquitos, P (in thousands), in an artificial lake in a housing estate is measured at the beginning of the year. The population after t months is given by the function, 
.
The rate of growth of the population is initially increasing. It then slows to be momentarily stationary in mid-winter (at t = 6), then continues to increase again in the last half of the year.
Determine the values of a and b.
Go to it.
Update
As Number 8 and Steve R hinted at and as Damo nailed, the central idiocy concerns the expression “the rate of population growth”, which means P'(t) and which then makes the problem unsolvable as written. Specifically:
- In the second paragraph, “it” has a stationary point of inflection when t = 6, which is impossible if “it” refers to the quadratic P'(t).
- On the other hand, if “it” refers to P(t) then solving gives a < 0. That implies P”(0) = 2a < 0, which means “the rate of population growth” (i.e. P’) is initially decreasing, contradicting the first claim of the second paragraph.
The most generous interpretation is that the examiners intended for the population P, not the rate P’, to be initially increasing. Other interpretations are less generous.
No matter the intent, the question is inexcusable. It is also worth noting that even if corrected the question is awful, a trivial inflection problem dressed up with idiotic modelling:
- Modelling population growth with a cubic is hilarious.
- Months is a pretty stupid unit of time.
- The
rate ofpopulationgrowthinitially increasing is irrelevant. - Why is the lake artificial? Who gives a shit?
- Why is the lake in a housing estate? Who gives a shit?
Finally, it’s “latter half” or “second half”, not “last half”. Yes, with all else awful here, it hardly matters. But it’s wrong.
Further Update
The marking schemes for the exam are now up, here and here. As was predicted, “the rate of growth of the population” was intended to mean “population”. As is predictable, the grading scheme gives no indication that the question is garbled garbage.
The gutless contempt with which certain educational authorities repeatedly treat students and teachers is a wonder to behold.
a=b=12 seems to work, but there are instantly problems with this answer before even beginning to look at the context.
Some issues include:
1. Does the “model” work to the extent all values for P(integer) are integers?
2. Is the only stationary point at t=6? This certainly seems to be the implication, but…
3. What is the “it” which slows, the population growth or the rate of population growth?
But there are undoubtedly others.
All relevant, but you missed the elephant …
Actually, I’ll amend that: you grazed the elephant.
Marty,
Not sure about the gorilla on the basketball court …
Or swimming mosquitoes ?
….But I don’t like the wording “initially increasing”
Setting the first two derivatives of P(6) to zero yields a=-18 and b = 108 but then P'(0) =108 and would then initially decrease as t increased above zero
Steve R
It is a bit gorilla-ish. I’m not sure I understand your concern with “initial decrease”.
If the question had said the population gradient P'(t) was initially decreasing above zero that would be fine ?
Ah, I see. So pointing to the same issue as Damo.
There is a problem with the wording in the second paragraph – I’m going to assume that they are not asking what they think they are asking. What gets me though is:
The population after t months is given by the function…
But months vary in length. Blargh.
Huh. I didn’t even see that. Blargh, indeed. Can you spell out what they are asking and what they think they are asking?
The rate of growth of the population is initially increasing.
This means that the second derivative would need to be positive. But the second derivative is 6t+2a. Which means that 2a would have to be positive, which means that it is impossible for the rate of growth to be stationary when t=6.
They think that they are asking:
The population is initially increasing…
And that would be the elephant.
I think there’s a little more awfulness in the question, but Damo and all have pointed out the main idiocy. I’ll update the post soon.
On a side note, I have learned to be suspicious of the wording “find the values of a and b” here in Victoria. In other parts of the world the question would be worded “find the value of a and the value of b” to avoid the question of whether there are multiple acceptable values of a and/or b that the examiner is later going to report that 99% of students missed (whereas actually 1449/1450 is not the same as 99%)
It is a very tricky thing to word questions in a clear and unambiguous manner. Of course, this doesn’t begin to excuse the garbage above.
Perhaps using a modified version of Gompertz Law (1825) would be a more sensible approach to modelling mosquito populations as there is an asymptotic limit when all the resources of the lake has been exhausted.
https://en.m.wikipedia.org/wiki/Gompertz_function
Thanks, Steve. I don’t know a thing about mathematical modelling, and had to click on your link to find out what a Gompertz function is. Real modelling is a difficult art, but of course the “modelling” in Methods is invariably garbage.
Confusing increasing rate of growth increasing versus the population increasing seems like a huge error. Like confusing second derivative with first. Or speed with acceleration.
Seems like exactly the type of error they wouldn’t want to make given all the fussing we routinely see with newspaper articles that confuse lower absolute spending versus lower rate of growth in spending. Or stuff like that.