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Math Help - Bacteria

  1. #1
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    Bacteria

    A certain strain of bacteria is living in a discarded Mountain Dew can. These little strappers are very good at reproducing and are capable of doubling their population every sixty seconds. Unfortunately, they only live for one hour. The first bacterium invaded the can at 10:00PM. At 10:57 after inhaling way too many Mountain Dew fumes, one bacterium is heard to shout, "Hooray! We've been alive for 57 whole minutes and the can is only 1/8 full. Party on Garth."

    a. How much longer will the party go on?

    As 11:00 approaches, an expedition is formed to locate a new place to live. A returning bacterium announces, "Hooray again! We have located 3 more empty Mountain Dew cans. That's three times more space than we've ever had before in our entire meager existence. We can probably go on reproducing forever. Party on Wayne"

    b. How much more time has the expedition provided?
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  2. #2
    Behold, the power of SARDINES!
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    Yee Ha

    So if we start with A(0)=A_0
    Then the number of bacteria at any time t is

    A(t) = \begin {cases}<br />
A_02^t \mbox{                 if          } 0 \le t < 60 \\<br />
A_02^t-A_02^{t-60} \mbox{ if } t \ge 60<br />
\end{cases}

    So if after 57 min the can is \frac{V}{8} full then

    A(57)=A_02^{57}=\frac{V}{8}

    So to fill the can we need

    8A_02^{57}=V \iff 2^{60}A_0=V

    Now we need to solve

    2^{60}A_0=A_02^t-A_02^{t-60}

    2^{60}A_0=A_0(2^t-2^t \cdot2^{-60}) \iff 2^{60}=2^t(1-\frac{1}{2^{60}})

    2^{60}=2^t \left(\frac{2^{60}-1}{2^{60}} \right)

    \frac{2^{120}}{2^{60}-1}=2^t \iff t= \frac{ln\frac{2^{120}}{2^{60}-1}}{ln2}

    You should be able to finish from here.

    Good luck.

    B.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by Nurseman View Post
    A certain strain of bacteria is living in a discarded Mountain Dew can. These little strappers are very good at reproducing and are capable of doubling their population every sixty seconds. Unfortunately, they only live for one hour. The first bacterium invaded the can at 10:00PM. At 10:57 after inhaling way too many Mountain Dew fumes, one bacterium is heard to shout, "Hooray! We've been alive for 57 whole minutes and the can is only 1/8 full. Party on Garth."

    a. How much longer will the party go on?

    As 11:00 approaches, an expedition is formed to locate a new place to live. A returning bacterium announces, "Hooray again! We have located 3 more empty Mountain Dew cans. That's three times more space than we've ever had before in our entire meager existence. We can probably go on reproducing forever. Party on Wayne"

    b. How much more time has the expedition provided?
    This is a poorly posed question. What does "How much longer will the
    party go on?" mean? Does it mean how long can they go on reporducing,
    or how long can they go on reproducucing at maximum rate, ...?

    RonL
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