Question

a homeowner is trying to reduce the insect population in the yard and needs to decide between setting up an insect zapper that zaps 15% of the insects every day or introducing dragonflies that will eat 100 insects every day. if the homeowner is having a backyard party in 4 days, which method will result in the fewest insects at the party?

Explanation:

Step1: Calculate insect - reduction for insect zapper

Let the initial number of insects be $x$. After the first day, the number of insects remaining with the insect - zapper is $x(1 - 0.15)$. After 4 days, if we start with $x$ insects, the number of insects remaining $N_{zapper}=x(1 - 0.15)^4=x\times0.85^4$.
$0.85^4=0.85\times0.85\times0.85\times0.85 = 0.52200625$.

Step2: Calculate insect - reduction for dragonflies

If dragonflies eat 100 insects per day, after 4 days, the number of insects eaten is $100\times4 = 400$. Let the initial number of insects be $x$. The number of insects remaining $N_{dragonflies}=x - 400$.

To compare which method results in the fewest insects, assume the initial number of insects is $x$.
If we want to find out when the zapper is better (fewer insects remaining), we set up the inequality $x\times0.85^4$x\times0.52200625$400$400 < 0.47799375x$.
$x>\frac{400}{0.47799375}\approx837$.

If the initial number of insects is large enough, the insect - zapper will result in fewer insects.

Let's assume we start with a large number of insects. The number of insects remaining with the insect - zapper method:
Let the initial number of insects be $I$. After 4 days, the number of insects $I_{zapper}=I\times0.85^4$.
After 4 days, with the dragonfly method, the number of insects $I_{dragonflies}=I - 400$.

If we assume $I = 1000$:
For the insect - zapper: $I_{zapper}=1000\times0.85^4=1000\times0.52200625 = 522.00625$.
For the dragonflies: $I_{dragonflies}=1000 - 400=600$.

Answer:

The insect - zapper will result in the fewest insects at the party.