Question

PLEASE ANSWER!! WILL MARK AS THE BRAINLIEST!! EXPLAIN IT TO ME BECAUSE I AM NOT UNDERSTANDING ANYTHING!!!!!

Answer 1

Short Answer D
The first thing that must be done is to see what e raised to a minus number does. Try a real simple example

y = 5e^-0.05
Put that into your calculator. You get
5 * 
e 
^ 
- 0.05
= 

The result is getting smaller. The answer I get is 4.7 which  is less than 5. That means that our insect population is going to go down. What will happen if you change that -0.05 to a plus 0.05 Try it again.

5*e^0.05 = I get 5.25 The number is increasing 5.25 > 5. 

What have you done? You have just eliminated two possible answers. a and c will both make 26 go up. You don't want either of them.

So now we need to look at B and D 
B won't work. Remember that t is a number in years. Any other number represents millions of insects. So the second equation must be written as P>2 The population must be kept at 2 million insects. Until then the first equation decreases the insect's numbers.

The answer is D

Answer 2

P represents the population in millions.

An exponential function with a positive exponent means function increases with time t.
An exponential function with a negative exponent means function decreases with time t.
So we're looking for a function with a NEGATIVE exponent.

Also, since question says population will remain CONSTANT at the minimum of 2 (millions) insects, so it should not be a function of time t.
Therefore this condition should be written as P>=2 (millions) [and not 2t].

Given the above  information, it will be up to you to choose the correct answer choice.