WitCH 5: What a West

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) 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, \color{blue}\boldsymbol{P(t) = t^3 + at^2 + bt + 2, 0\leqslant t \leqslant 12}.

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 of population growth initially 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.