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.

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

General Exponential function

f(x) = A*b^t

where A = initial amount, when t=0 and b is the rate of change or growth for increasing time.

Answer 3

since it grows wildly with no competition, the population , i assume, can go to infinity here.
It is not a logistic problem. Just a general exponential function.

Answer 4

Recall:

\[[a]^{2x} = [a^2]^x\]

Do that for the second question

Answer 5

so would that be (206^2)^x?

Answer 6

No , 200 is the initial population at time t=0. (1.06)^(2x) is the growth rate.

\[g(x) = 200*[1.03]^{2x} = 200*[1.03^2]^x = 200*[1.0609]^x\]

You don't square the initial population.

Answer 7

If x represents the number of years since time t=0. Then each increase of 1 year, multiplies the initial population, 200, by 1.0609. The population will increase by 6.09% each year from last years population.

Answer 8

should i also write what u r saying or is it an explanation for me

Answer 9

Not sure if they want you to make up numbers for the first invasive population, for A and b in.

f(x) = A*b^x

Answer 10

Invasive Population: f(x) = A*b^x
local population : g(x) = 200 *(1.0609)^x
--------------------------------------
compare the 2, initial populations 200 vs A, and growth rate each year of (1.0609) vs b

Answer 11

For the invasive Populations Graph, The y intercept is where X=0, that is the value A in
f(x) = A*b^x

The domain is all real numbers greater than time x=0 , the range is from the initial population A to infinity population.

Answer 12

Average Rate of change for the population will represent the slope of a secant line connecting the points (2,f(2)) and (5,f(5)) on the graph of population vs time. It would give you the average growth of the population per year over those years. It is found by calculating.
\[Ave~growth~=\frac{ f(5)-f(2) }{ 5-2 }\]

Answer 13

had to get it all down, it should all be there to help ya... got to run . ttyl

Answer 14

Thankssss so much :D