anonymous
  • anonymous
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.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
@whpalmer4, anyone?
anonymous
  • anonymous
@sammixboo
anonymous
  • anonymous
@prowrestler

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anonymous
  • anonymous
@Michele_Laino please please help me or ask someone for me
Michele_Laino
  • Michele_Laino
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: \[\Large \begin{gathered} f\left( x \right) = {3^x} \hfill \\ g\left( x \right) = 5x - 1 \hfill \\ \end{gathered} \]
Michele_Laino
  • Michele_Laino
then I draw the graph of both functions f(x) and g(x) and I search for intersection point of those graphs
anonymous
  • anonymous
is there a way of finding the intersection without graphing
Michele_Laino
  • Michele_Laino
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: \[\Large \left( {2,9} \right)\]
Michele_Laino
  • Michele_Laino
another point can be compute, if we expand the function f(x) around x=0, using Taylor expansion
Michele_Laino
  • Michele_Laino
computed*
anonymous
  • anonymous
where did you get 2 from
Michele_Laino
  • Michele_Laino
I did some trial
anonymous
  • anonymous
oh ok and thank you
Michele_Laino
  • Michele_Laino
please wait, try to write the Taylor expansion, around, x=0 of f(x), or try to use a software online like "desmos"
anonymous
  • anonymous
hold the question says use system of equations
Michele_Laino
  • Michele_Laino
yes! in fact I broke your equation in two functions
Michele_Laino
  • Michele_Laino
here is the system: \[\Large \left\{ \begin{gathered} f\left( x \right) = {3^x} \hfill \\ g\left( x \right) = 5x - 1 \hfill \\ \end{gathered} \right.\]
anonymous
  • anonymous
but there is no way of eliminating anything so is that why we graph
Michele_Laino
  • Michele_Laino
for example, I write the Taylor expansion of f(x):
anonymous
  • anonymous
ok
anonymous
  • anonymous
it is just that i did not learn about the taylor expansion
Michele_Laino
  • Michele_Laino
please here is the expansion up to the first order term:
Michele_Laino
  • Michele_Laino
\[\Large {3^x} \simeq {\left. {{3^x}} \right|_{x = 0}} + {\left. {{3^x}\log 3} \right|_{x = 0}}x = 1 + x\log 3\]
Michele_Laino
  • Michele_Laino
now, if we want to solve your equation, namely: 3^x=5x-1, we can solve this equation: \[\Large 1 + x\log 3 = 5x - 1\]
Michele_Laino
  • Michele_Laino
so we get the second intersection point
anonymous
  • anonymous
ok thank you, no need to go further.
Michele_Laino
  • Michele_Laino
:)

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