I don't watch TV or do anything else when I study. I demand absolute silence... other than the sound of my air conditioner (if I'm in Singapore). You can ask my sister about how particular I am about this...

Anyway, even in the answers of textbooks and stuff, the flooring and ceiling functions aren't used consistently for questions like in the year and related questions; I've been in correspondence with some textbook authors, and supposedly there's no real consensus...

Sorry for the late solution, I was on a self imposed maths exile for 3 months.

]]>Students, we're now in a grave situation.

Thanks to calculations provided by my Maths HOD

Someone, I now knowfrom the startthat the number of infected cases will breach 45% int= 1.78 hours' time at 6:47pm. Thus I'm able to schedule this announcement now at 6 pm to warn everyone in advancebeforeit happens.A careless student (who doesn't read the question properly) may have simply

rounded up1.78 to 2 hours and erroneously schedule my announcement at 8 pm. By then I might only be able to "Oink oink oink" through the announcement!Therefore it's vital for all of you to read such questions on

application of exponential or logarithmic functionscarefully and understand its context before submitting your final answer in your exam.For e.g. some ask for "

after how many years..." vs "find the year in which..." - this is especially tricky with typical 'population' questions where you're asked for e.g. "the yearin whichthe population first hits belowx...".Do have a look at this forum entry (where my idol Miss Loi have commented) for further details.

I HEREBY DECLARE THE SCHOOL CLOSED FOR SEVEN DAYS!

*rapturous applause* *throws confetti*

]]>

→ Yes the numbers **3, 9, 27** at the base are simply crying their hearts out to you to express them in terms of the powers of the common factor 3, in order to 'bring down' the unknown `α` that's helplessly trapped within the indices (via comparing the indices on both sides).

→ Recall that for the *same base*, Multiply is to Add: `a`^{m} × `a`^{n} = `a`^{m+n} and Power is to Multiply: (`a`^{m})^{n} = `a`^{mn}

→ Some students will commit here the usual terrible sin of 2 lg (`α` − 20) = 2(lg `α` − lg 20) **(WRONG!)**. Recall your **Product Law of Logarithms**: log_{a}(`xy`) *then can* log_{a}`x` + log_{a}`y`!

→ Recall as well the **Power Law of Logarithms**: `r` log_{a}`x` = log_{a}`x`^{r} that's used here to prepare the lg term on the RHS for the Quotient Law later.

We've successfully reduced the bases to a single 3 on both sides - now we can "bring down" the lg terms by comparing the indices on both sides!

→ Another possible spot for a terrible sin - some students will do a lg `α` − lg (`α` − 20)^{2} = lg (`α` − (`α` − 20)^{2} ) - **WRONG!** Recall your **Quotient Law of Logarithms**: log_{a}`x` − log_{a}`y` = log_{a}()!

→ sadly there're still those who can get stuck at this stage not knowing how to "un-lg/log/ln" their logarithmic/exponential equations. Recall that:

'Un-log': log_{a} `y` = `x` ⇔ `y` = `a`^{x}

'Un-lg': lg `y` = `x` ⇔ `y` = 10^{x}

'Un-ln': ln `y` = `x` ⇔ `y` = e^{x}

→ Those of you who revise while watching TV may miss out the all-important property of:

log_{a}`y` is defined iff `y` > 0 and `a` > 0

.. and get stuck at this point not knowing which value of `α` to pick from the resulting quadratic equation (*Tsk tsk*.)

reject since

Well and truly written by **Someone** who obviously doesn't watch TV while revising (Yes?) 😉

With this, we've obtained `α` = 40 from this beautiful logarithmic equation that requires the entire arsenal of Logarithmic weaponry i.e. basic logarithm properties and the Product Law, Quotient Law, Power Law, as well as the Rules of Indices. Wicked isn't it? 😉

→ So now we move into the first (and typically more straightforward) part of the question, which is simply substituting `α`=40 and`t`=16 into the equation. You'll find that e^{−0.1(40)(16)} is very, very small (≈0) hence i.e. the entire school has been infected by the 16th hour.

→ In Part 2, we're finding the value of `t` when number of infected students crosses the 45% mark. So we begin Part 2 by substituting `y` = 45% × 1500 = 675 into the equation to obtain `t` ...

→ and this is the part where you 'ln both sides' to get `t` out from that e ...

→ and viola! We've found that `t`=1.78 when 45% of the students gets infected. But this is also the point where many gets a little infected by fatigue after such long and hard calculations that they didn't **read the question properly** when presenting the final answer. See the Principal's inspiring speech at the next comment...

**Babyface:** *Dials 993* *sweats*