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GalacticPlunder

  • one year ago

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.

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  1. GalacticPlunder
    • one year ago
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    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.

  2. GalacticPlunder
    • one year ago
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    2. 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.

  3. GalacticPlunder
    • one year ago
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    3. 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.

  4. GalacticPlunder
    • one year ago
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    4. 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.

  5. SolomonZelman
    • one year ago
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    EZ! you just need to make an exponential function up - any exponential function you like. An exponential function (abstractly) is in a form of \(\large\color{black}{ \displaystyle y=a(b)^x }\) I will assume you know: \(\large\color{black}{ \displaystyle {\rm C}^0=1 }\) (for any non-zero number C)

  6. SolomonZelman
    • one year ago
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    \(\LARGE\color{black}{ \displaystyle \color{darkgoldenrod}{y}=\color{red}{a}(\color{blue}{b})^{\color{green}{x}} }\) \(\normalsize \color{black}{ \displaystyle \color{darkgoldenrod}{\rm Number~of~snalis}=\color{red}{\rm Initial~population}\cdot (\color{blue}{\rm growth~rate})^{\color{green}{\large ~\rm \text{#}~of~years }} }\)

  7. GalacticPlunder
    • one year ago
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    So I just input random numbers?

  8. SolomonZelman
    • one year ago
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    yes, but a must be positive, and b>1

  9. SolomonZelman
    • one year ago
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    So that you don't end up getting a negative population.... and I would advise to choose Natural numbers for a and b.

  10. SolomonZelman
    • one year ago
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    natural numbers are: 1 , 2, 3, 4, 5, 6, 7, etc....

  11. GalacticPlunder
    • one year ago
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    I'm still kind of stuck.

  12. SolomonZelman
    • one year ago
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    try to put random values for a and b (once)

  13. GalacticPlunder
    • one year ago
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    Would y=10, a=5, b=2, x=2 work?

  14. SolomonZelman
    • one year ago
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    you don't need to plug in anything for x and y.

  15. SolomonZelman
    • one year ago
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    So you created a function: \(\large\color{black}{ \displaystyle y=5(2)^x }\)

  16. SolomonZelman
    • one year ago
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    Now lets explain to Iris (:O) what is what....

  17. GalacticPlunder
    • one year ago
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    Ok... so how would I explain it?

  18. GalacticPlunder
    • one year ago
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    Y= Number of snails a=Initial population⋅ b=growth rate x=number of years ?

  19. SolomonZelman
    • one year ago
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    Lets do that together. You need to explain what and why is the principal (initial) population of snails & what and why is the rate of growth of the population.

  20. SolomonZelman
    • one year ago
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    Ok, will you agree that the initial population is at year zero, or in other words when x=0?

  21. GalacticPlunder
    • one year ago
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    Ok. So that would mean a = 0?

  22. SolomonZelman
    • one year ago
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    no

  23. GalacticPlunder
    • one year ago
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    When x = 0

  24. SolomonZelman
    • one year ago
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    \(\large\color{black}{ \displaystyle y=5(2)^x }\) Initially (at year zero - i.e. at x=0) \(\large\color{black}{ \displaystyle y=5(2)^0 }\) \(\large\color{black}{ \displaystyle y=5(1) }\) \(\large\color{black}{ \displaystyle y=5 }\)

  25. SolomonZelman
    • one year ago
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    See why the initial/principal population (when x=0) is (equivalent to) 5 (snails) ?

  26. GalacticPlunder
    • one year ago
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    So there was no increase?

  27. SolomonZelman
    • one year ago
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    There is going to be an increase as years pass by, BUT the population didn't increase right away when started.

  28. SolomonZelman
    • one year ago
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    This that we are discussing right now is just the principal population - or from what number of snails did we start.

  29. GalacticPlunder
    • one year ago
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    Oh ok. I get it now. So what would be the starting number? And how would I explain my function?

  30. SolomonZelman
    • one year ago
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    I will be gone. tag someone else. (I am constructing my basement and bath)

  31. SolomonZelman
    • one year ago
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    sorry

  32. GalacticPlunder
    • one year ago
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    Ok thanks for helping.

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