Question

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.

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.

Answer 1

\[g(x) = 200(1.03)^{2x}=200[(1.03)^2]^x\] so first thing you want to do is square \(1.03\) i.e. compute
\[(1.03)(1.03)\]with a calculator
that will give you the answer to the first part

Answer 2

we are looking at part 1 right? the part that says

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.

Answer 3

so the first thing you want to do as write it as a function where the exponent is \(x\) instead of \(2x\)
clear?

Answer 4

in other words, instead of
\[ g(x) = 200(1.03)^{2x}\] you want
\[g(x)=200\times b^x\]for some number \(b\) instead of \(1.03\)
your job is to find what \(b\) is

Answer 5

do you understand that you need to find \(b\)? i am not asking if you know what \(b\) is, only if you know that is what you need to find

Answer 6

ok
now by the laws of exponents,
\[(1.03)^{2x}=((1.03)^2)^x\]

Answer 7

so in order to find the \(b\) what you need to do is find \((1.03)^2=(1.03)(1.03)\) i would use a calculator

Answer 8

yes

Answer 9

let me know what you get

Answer 10

use a calculator

Answer 11

close
did you use a calculator?

Answer 12

then make sure to write the answer correctly
it is not \(1.069\)

Answer 13

ok good
so now we know
\[200(1.03)^{2x}=200(1.0609)^x\]

Answer 14

that accomplishes job 1, write with an \(x\) in the exponent
lol yes, we only did one thing so far... multiply \((1.03)\times (1.03)\)

Answer 15

i thought we were doing this
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.

Answer 16

we are, the thing on top is just information about iris.

Answer 17

the first one says to make up your exponential function for a snail poplulation
lets do that quickly
how many snails would you like to start with?

Answer 18

ooh, no you did the second part.

Answer 19

the first part opf the problem is 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.

Answer 20

uhm..... 5 snails i guess.

Answer 21

ok fine
what percent would you like the population to increase per year?

Answer 22

5%

Answer 23

ok good
now we have almost answered question one completely

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.

a) the "principle population is 5"
b) the growth rate per year is \(5\%\)
c) the function \(f(x)\) will be \(f(x)=5(1.05)^x\)

Answer 24

in complete sentences, the principle population is 5 which is why out front there is a 5 in
\[f(x)=\color{red}5(1.05)^x\]

Answer 25

the growth rate per year is \(5\%=.05\)
to increase a number by \(5\%\) you multiply it by \(100\%+5\%=105\%=1.05\)

Answer 26

to do it again and again for \(x\) years you multiply by \((1.05)^x\) which is why the function is
\[f(x)=5(1.05)^x\] and now we are done with that part completely

Answer 27

so, for part one for the ansswer i put
100%+5%=105%=1.05
f(x)=5(1.05)x

Answer 28

ok

Answer 29

im asking you

Answer 30

it asked for "complete sentences"
i wrote a couple above, but you can write them in your own words

Answer 31

hold on im gonna make mmy own sentences and then say then to you and i want you to make sure it sounds right is that ok?

Answer 32

sure
but i gotta run in ten minutes, so be brief

Answer 33

well since you gatta gho soon lets just move on, because i really want to finish this.

Answer 34

do we need to redo part 2?

Answer 35

no that part is good
\[200(1.03)^{2x}=200(1.0609)^x\] because
\[(1.03)(1.03)=1.0609\]

Answer 36

so third part.t

Answer 37

as for
Then explain to Iris how the key features of this local snail population compares to the key features of the invasive population.
i would say that comparing
\[g(x)=200(1.0609)^x\] so
\[f(x)=5(1.05)^x\] the first one starts with 200 and grows at a rate of \(6.09\%\) per year, whereas the second one starts with 5 and grows at a rate of \(5\%\) per year

Answer 38

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.

Answer 39

domain, since \(x\)represents time, we would say that the domain would be \(x>0\) since you can't go backwards in time

Answer 40

range would be \(y\geq 5\) since we started with 5

Answer 41

the \(y\)intercept is how many snails you started with
you said 5, so the y intercept is 5

Answer 42

and the function is increasing, because the population is getting bigger, not smaller

Answer 43

im confused on what i should write for part three though.

Answer 44

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 45

i take it we are using
\[f(x)=5(1.05)^x\]for this one, right?

Answer 46

wait, im still stuck on three.

Answer 47

all i have for three is what the domain and range are.

Answer 48

that is what it asks for

Answer 49

?

Answer 50

oh it also asks if it is increasing
the population is growing, so it is increasing

Answer 51

thats all it asks for ?

Answer 52

alright last part.

Answer 53

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 54

if we are using \(f(x)=5(1.05)^x\) in year 2 there will be \(5(1.05)^2\)snails and in year 5 there will be \(1.05)^5\) snails
compute these numbers first

Answer 55

then the average is \[5(1.05)^5-5(1.05)^2\]divided by the number of years, which is 3, since \(5-2=3\)

Answer 56

hey sorry my internet cut off

Answer 57

uhm what would tyou like me to do my computer cut me off, and im lost now.

Answer 58

so 5(1.05)2 you want me to solve and 5(1.05)5?

Answer 59

your messages keep disapearing.

Answer 60

for 5(1.05)2 the answer i got for that is 27.5625

Answer 61

@satellite73