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anonymous

  • one year ago

A biologist is comparing the growth of a population of flies per week to the number of flies a lizard will consume per week. She has devised an equation to solve for which day (x) the lizard would be able to eat the entire population. The equation is 3^x = 5x − 1. Explain to the biologist how she can solve this on a graph using a system of equations.

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  1. anonymous
    • one year ago
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    @whpalmer4, anyone?

  2. anonymous
    • one year ago
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    @sammixboo

  3. anonymous
    • one year ago
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    @prowrestler

  4. anonymous
    • one year ago
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    @Michele_Laino please please help me or ask someone for me

  5. Michele_Laino
    • one year ago
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    A possible way is: I call with g(x) the function 3^x, and with g(x) the function 5x-1, namely I write this: \[\Large \begin{gathered} f\left( x \right) = {3^x} \hfill \\ g\left( x \right) = 5x - 1 \hfill \\ \end{gathered} \]

  6. Michele_Laino
    • one year ago
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    then I draw the graph of both functions f(x) and g(x) and I search for intersection point of those graphs

  7. anonymous
    • one year ago
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    is there a way of finding the intersection without graphing

  8. Michele_Laino
    • one year ago
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    one point is given setting x=2 we have: \[\Large \begin{gathered} f\left( 2 \right) = {3^2} = 9 \hfill \\ g\left( 2 \right) = 5 \cdot 2 - 1 = 9 \hfill \\ \end{gathered} \] so the corresponding intersection point is: \[\Large \left( {2,9} \right)\]

  9. Michele_Laino
    • one year ago
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    another point can be compute, if we expand the function f(x) around x=0, using Taylor expansion

  10. Michele_Laino
    • one year ago
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    computed*

  11. anonymous
    • one year ago
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    where did you get 2 from

  12. Michele_Laino
    • one year ago
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    I did some trial

  13. anonymous
    • one year ago
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    oh ok and thank you

  14. Michele_Laino
    • one year ago
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    please wait, try to write the Taylor expansion, around, x=0 of f(x), or try to use a software online like "desmos"

  15. anonymous
    • one year ago
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    hold the question says use system of equations

  16. Michele_Laino
    • one year ago
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    yes! in fact I broke your equation in two functions

  17. Michele_Laino
    • one year ago
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    here is the system: \[\Large \left\{ \begin{gathered} f\left( x \right) = {3^x} \hfill \\ g\left( x \right) = 5x - 1 \hfill \\ \end{gathered} \right.\]

  18. anonymous
    • one year ago
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    but there is no way of eliminating anything so is that why we graph

  19. Michele_Laino
    • one year ago
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    for example, I write the Taylor expansion of f(x):

  20. anonymous
    • one year ago
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    ok

  21. anonymous
    • one year ago
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    it is just that i did not learn about the taylor expansion

  22. Michele_Laino
    • one year ago
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    please here is the expansion up to the first order term:

  23. Michele_Laino
    • one year ago
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    \[\Large {3^x} \simeq {\left. {{3^x}} \right|_{x = 0}} + {\left. {{3^x}\log 3} \right|_{x = 0}}x = 1 + x\log 3\]

  24. Michele_Laino
    • one year ago
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    now, if we want to solve your equation, namely: 3^x=5x-1, we can solve this equation: \[\Large 1 + x\log 3 = 5x - 1\]

  25. Michele_Laino
    • one year ago
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    so we get the second intersection point

  26. anonymous
    • one year ago
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    ok thank you, no need to go further.

  27. Michele_Laino
    • one year ago
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    :)

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