Question

1.Assume that 20 squirrels are put in an enclosed wildlife ranch and the squirrel population grows for the next 5 years, as shown in the table.

Year

Rabbit Population

0

20

1

80

2

320

3

1280

4

5120

5

20,480
Write a function that represents this.













2.A second group of 20 squirrels of a different breed is placed on a different ranch. The population of squirrels triples each year. Write an equation to model this situation.


















3.Had the second group started with 40 squirrels instead of 20, which population will be bigger by year 3 and by how much?

Answer 1

For the first group we have the pairs:

year population

0 20

1 80

2 320

3 1,280

4 5,120

5 20,480

Here we can see that the population quadruples each year

(4*20 = 80, 80*4 = 320, 320*4 = 1,280, etc...)

then the population equation is:

P(0) = 20

P(1) = 20*4

P(2) = (20*4)*4 = 20*4^2

We already can see the pattern, then we can write this relationship as:

P(t) = A*(4)^(t)

Where:

t represents time in years, and A is the initial population, that we know it is 20, then:

P(t) = 20*(4)^t

This is the function that represents the table.

B) Now we have a group of 20 squirrels and the population triples each year, with the same reasoning than before we can write the equation that models this situation as:

Q(t) = 20*(3)^t

C) Now, if the initial population of the second group is 40, the equation becomes:

Q(t) = 40*(3)^t

The population by year 3 is given by replacing t by 3, then:

Q(3) = 40*(3)^3 = 1080

And the population of the other group in year 3 is seen in the table, it is 1,280, then the population of the first group is bigger by year 3, and it is greater by:

1,280 - 1,080 = 200

So the first group is larger by 200 squirrels.