The population of Summertown varies according to the model $$ S(t)=-1200\cos\left(\frac { \pi }{ 6 } t\right)+1500 $$ The population of Cool Ridge varies according to the model $$ C(t)=1200(1.025)^{ t } $$ In both models, $t$ represents the number of months since January 1, 2004. Based on these models, how many times after January 1, 2004, will $S(t)=C(t)$?
Is there a quick way to find out without using a graphing calculator?
First show that the function $C(t)$ is strictly increasing from $C(0)=1200$.
Then the function $S(t)$ is sinusoidal with minimum at $300$, maximum at $2700$ and period 12. That means that it's strictly increasing on $(0,6)$ then strictly decreasing on $(6,12)$ and so on.
On each of these intervals, by the Intermediate value theorem, the function $S(t)-C(t)$ must be equal to $0$ exactly once, until $C(t)$ grows too big (larger than the max of $S(t)$ i.e $2700$).
You need the value of $t$ such that $1200(1.025)^t=2700$, this is given by \begin{align} (41/40)^t&=\frac{2700}{1200}=\frac{9}{4}\\ t &= \frac{\ln(9/4)}{\ln(41/40)}\\ \end{align} If you have access to a basic calculation, this yields $t\approx 32.8$
Therefore we will have $C(t)=S(t)$ exactly once on each of the intervals $(0,6)$, $(6,12)$, $(12,18)$, $(18,24)$, $(24,30)$ and $(30,36)$, hence 6 times.
Edit Note that this works because the last interval on $(30,36)$ is decreasing, otherwise we would have to look at in details.