Question

Algebra 1 question please help!
Iris has been studying an invasive population of snails. This particular snail has no local predators, so the population grows wildly. She has observed that the population follows an exponential rate of growth for fifteen years.

Answer 1

Create your own exponential function, f(x), which models the snail population. You will need to identify the principal population of the snails and the rate of growth each year. Explain to Iris how your function shows the principal population and the rate of growth, in complete sentences.


A local snail population grows according to the function g(x) = 200(1.03)2x. Demonstrate the steps to convert g(x) into an equivalent function with only x as the exponent. Then, explain to Iris how the key features of this local snail population compares to the key features of the invasive population.


Iris wants to graph the invasive snail population to show the city council. Justify what the appropriate domain and range would be for the function f(x), what the y-intercept would be, and if the function is increasing or decreasing.


In five years, a garden festival plans on using the park where Iris has been studying the invasive snails. Explain to the garden festival committee how to find the average rate of change for the snail population between years 2 and 5. Describe what this average rate of change represents.

Answer 2

You have a ton of instructions. What's preventing you from following them?

Answer 3

not knowing how to do them :(

Answer 4

Are you in the right class?

An answer to the first paragraph is given in the 2nd paragraph. Do you see it?

Answer 5

Is that supposed to be \(g(x) = 200\cdot (1 + 0.03)^{2x}\) or just \(g(x) = 200\cdot (1 + 0.03)^{x}\)?

Do you see the "Initial Population"? Try g(0) and see what happens.
Do you see the "growth rate"? I highlighted it by separating it inside the parentheses.

Answer 6

g(0)=200(1+.03)^2(0)
200+6
206

Answer 7

is that right?

Answer 8

Not so good. 1.03^(2*0) = 1.03^0 = 1, leaving just 200. This is the initial population.

Okay, we have a little problem. We are given \(g(x) = 200\cdot(1.03)^{2x}\). This is a little annoying because that exponent has the '2' in it. We do need to get rid of that if we wish to know the true annual growth rate. Let's try this:

\(g(x) = 200\cdot(1.03)^{2x} = 200\cdot\left((1.03)^{2}\right)^{x} = 200\cdot (1.0609)^{x}\)

You need to see how that happened. We had a "0.03" or a 3% growth rate staring at us, but with that 2 in the exponent, it was really a 6.09% annual growth rate. Do you see that this is so?

Answer 9

yes i do

Answer 10

Okay, now we have to back up a little. We cheated when we grabbed g(x) as our invasive population. That's no good, as g(x) is really the native population. If we are going to create an invasive population, it will need a growth rate greater than the native population. Do you agree that this is so?

Answer 11

i agree

Answer 12

Okay, how about \(I(x)\) for the invasive species?

Pick a beginning population?
Pick a growth rate.

Anything, really. Just pick some.

Answer 13

10, sorry my computer froze

Answer 14

a growth rate of 5%

Answer 15

5% won't do it. We have to get over 6.09% Remember that?

Answer 16

oh yeah, 10%

Answer 17

Can you now write all of \(I(x)\)? Make it look just about like g(x).

Answer 18

I(x)=10(5)

Answer 19

im not confident thats correct.

Answer 20

No good. \(I(x) = 10\cdot (1.05)^{x}\) -- Look more familiar?

Answer 21

yes it does

Answer 22

Just to make a point, \(10(5) = 10(1+4)\) -- And that's a 400% growth rate!!! Not quite what we had in mind, is it?

Okay, 5% is still too small. Let's just go with 10%, okay? \(I(x) = 10\cdot(1+0.10)^{x}\).

Does that look invasive?

Answer 23

yeah my mistake lol, and yes

Answer 24

Did you notice that while we were talking we completed the first two parts of the assignment? :-)

Answer 25

i noticed we completed the first part but just now noticed we completed the second part, i like this style of teaching lol

Answer 26

Just kind of sneaking it in there, eh?

How about part 3? Domain. This is the same as "When do we care"? What do you think? Do we care about anything BEFORE time = 0?

Answer 27

no lol

Answer 28

domain represents time and range is the population right?

Answer 29

Just to comment on it.

For the native population, there MAY be some interest prior to time zero, since the population didn't just pop into existence.
For the invasive population, it simply did not exist prior to time zero. There can't be much interest at Time = -2!!

Domain: \(x \ge 0\)

y- intercepts. This is easy. Just evaluate \(g(0)\;and\;I(0)\). What do you get? You tried \(g(0)\) before and it didn't go so well. Do better this time. :-)

Answer 30

i hope i got it right this time

Answer 31

And g(0)?

Answer 32

Perfect. The next part is actually a little silly.

Are these functions increasing, decreasing, or neither, or one of each, or what?

Answer 33

if the function is g(0) or I(0) then its neither if its above 0 then its increasing

Answer 34

?? I don't understand that. We designed \(I(x)\) so that it would have a greater GROWTH rate than \(g(x)\). Are we saying ANYTHING about ANYTHING except GROWTH? I think not! These are strictly increasing. Make sense?

Answer 35

yeah it does

Answer 36

See, I told you it was silly. Everything is growing. Why would it be anything else?

Okay, now the last part. This is trickier. I'll copy the paragraph, here.

In five years, a garden festival plans on using the park where Iris has been studying the invasive snails. Explain to the garden festival committee how to find the average rate of change for the snail population between years 2 and 5. Describe what this average rate of change represents.

What do you think? How can we find the average rate of change from year 2 to year 5?

Consider \(g(2)\;and\;g(5)\).
Consider \(I(2)\;and\;I(5)\).

Answer 37

is that correct so far?

Answer 38

Yup. Good work. Now for \(I(x)\)...

Answer 39

Gaa!!! You switched back to 5%, again! Go make that 10% and take another shot at it. :-)

Now, we have another odd item that we have to clear up. It asks for "Average Growth RATE". Well, the growth rate for both \(g(x)\;and\;I(x)\) never changes. It's 6.09% and 10%. So, is the question asking anything? It's not perfectly clear.

Anyway,

It may want this, \(\dfrac{g(5) - g(2)}{3}\). That is an average GROWTH, not an average growth RATE, but it may be what is wanted.

It may want this, \(\dfrac{g(5) - g(2)}{3g(2)}\). That is sort of an average growth RATE, but a little odd. Still, it may be what is wanted.

I can't settle this without talking to the question's author.

Okay, well, I need to go. Calculate those values and think about them. That's all we can do for now.

Answer 40

alright thanks for all your help! and alright ill give it another shot. i think i can finish the problem now. thanks again! :)