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mr.luna

  • 3 years ago

how do i set this up?! a bacteria culture initially contains 100 dells and grows at a rate proportional to its size. after an hour the population has incresed to 420. a)find an expression for the number of bacteria after t hours. b)find the number of bacteria after 3 hours c)find the rate of growth after 3 hours d)when will the population reach 10,000

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  1. mr.luna
    • 3 years ago
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    i have trouble converting it to numbers. here what i have so far and i working on more. i use the P(t)=P(0)e^kt

  2. mr.luna
    • 3 years ago
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    so P(0)=100 P(3)=420

  3. mr.luna
    • 3 years ago
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    i set the equation P(3)=100e^3k=420 i solved and got k=ln42/3

  4. mr.luna
    • 3 years ago
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    is a [100e^[(ln42/3)(t)]

  5. mr.luna
    • 3 years ago
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    b) i just plug in 3 for t to the equation in a) ?

  6. mr.luna
    • 3 years ago
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    for d) i put equation a) equal to 10,000 right?

  7. mr.luna
    • 3 years ago
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    im stuck on c) though. how do you do that?

  8. victorarana
    • 3 years ago
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    I guess that you can derive the expression you got before to obtain dP/dt. And then you sustitute the value of 3 on it. .

  9. mr.luna
    • 3 years ago
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    i got it down to 100(42^t/3) i dont how to dervie?

  10. victorarana
    • 3 years ago
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    I think your initial conditions are wrong. You wrote P(3) = 420, and on the problem says P(1) = 420

  11. victorarana
    • 3 years ago
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    \[420 = 100e^{k(1)}\]

  12. victorarana
    • 3 years ago
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    so, \[k = \ln4.2\]

  13. mr.luna
    • 3 years ago
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    ok crap .yeah your right! good catch

  14. victorarana
    • 3 years ago
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    \[P = 100e^{\ln(4.2)t}\]

  15. victorarana
    • 3 years ago
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    if you derive you get: \[\frac{d}{dt}100e^{(\ln 4.2)k}= 100(\ln4.2)e^{(\ln 4.2)k}\]

  16. victorarana
    • 3 years ago
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    sorry its t instead of k

  17. mr.luna
    • 3 years ago
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    im trying to get waht you got for the derviative right now. its not coming out for me.

  18. victorarana
    • 3 years ago
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    what did you get?

  19. mr.luna
    • 3 years ago
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    instead of ln4.2 i get 4.2k

  20. victorarana
    • 3 years ago
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    remember that \[\frac{d}{dx}e^{ax} = ae^x\]

  21. victorarana
    • 3 years ago
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    Sorry i missed an "a" \[\frac{d}{dx}e^{ax}=ae^{ax}\]

  22. mr.luna
    • 3 years ago
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    oh. ok. got it now. thanks

  23. victorarana
    • 3 years ago
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    You're welcome.

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