Vtyjkyuk

An insect pest population doubles every 18 days. If an insecticide kills 90% of the insects, how often should it be applied to keep in insect population in check?

I tried using this exponential growth/decay model: $$A = Pleft(fracA_1P_1right)^t/t_1$$
where $P_1$ (initial value), $A_1$ (new value), $t$ (time period).

However, would this be the proper way to do it? I'm not quite sure how to interpret (and thus solve) this problem.

So you have an insect population that doubles every 18 days. Let's call it $m_0$. After 18 days, you will have $2m_0$, after another 18 days, you will have $4m_0$. The idea is that you model this growth with a function:

$f(t) = textsomething$,

but we know some values of this function:

$f(0 textdays) = m_0,$

$f(18 textdays) = 2m_0,$

$f(36 textdays) = 4m_0,$

$f(54 textdays) = 8m_0,$ and so on.

We know the form of the function:

$f(t textdays) = m_0 cdot n^t.$

To find $n$, we do:

$f(18 textdays) = 2m_0 = m_0 cdot n^18 textdays,$

$n=2^1/18 textdays.$

Our function is:

$f(t textdays) = m_0 cdot 2^t/18 textdays.$

Remember that the insect population is always growing, and it will hit the desired number of insects (or a critical number of insects) at $t_c$ (critical time). Let's call that critical number of insects $m_c$. Then you apply the insecticide and now the insect population is:

$$f(t_c)=m_0 cdot 2^t_c/18 = 0.1m_c$$

Then you have a new function of insect population:

$f(t textdays) = 0.1m_c cdot 2^t/18 textdays.$

Since the question is "how often should it be applied to keep the insect population in check?", the insect population will grow and it will hit the desired number of insects, $m_c$ after some time $t$, we want to know that time (in days):

$0.1m_c cdot 2^t/18 = m_c,$

and the result is (solve for $t$!):

$t=59.79 textdays.$