Results 1 to 4 of 4

Math Help - exponential functions

  1. #1
    Junior Member
    Joined
    Sep 2007
    Posts
    37

    exponential functions

    okay, this is a weird problem...thanks to anyone who can solve it

    A certain strain of bacteria is living in a can. They are good at reproducing and are capable of doubling their population every 60 seconds. Unfortunately, they only live for 1 hour. The first bacterium invaded the can at 10:00 PM and at 10:57 PM one bacterium shouts, "We've been alive for 57 minutes and the can is only one-eighth full. Party."
    --A--How much longer will the party go on?

    As 11:00 approaches, an expedition is formed to locate a new can to live. A returning bacterium announces, "Hooray, we have located 3 more empty cans. Thats three times more space than we've ever had in our entire existence.
    --B--How much more time has the expedition provided?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,830
    Thanks
    123
    Quote Originally Posted by doctorgk View Post
    okay, this is a weird problem...thanks to anyone who can solve it

    A certain strain of bacteria is living in a can. They are good at reproducing and are capable of doubling their population every 60 seconds. Unfortunately, they only live for 1 hour. The first bacterium invaded the can at 10:00 PM and at 10:57 PM one bacterium shouts, "We've been alive for 57 minutes and the can is only one-eighth full. Party."
    --A--How much longer will the party go on?

    ...
    If the time is measured in minutes the amount of bacteria can be calculated by:

    a(t) = 1\cdot 2^t

    After 60 minutes the first bacterium will die. There are missing 3 minutes to this moment. Since 2 = 8 the population of bacteria will increase 8 times and will fill the can completely!

    ... and the party will become a funeral
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,830
    Thanks
    123
    Quote Originally Posted by doctorgk View Post
    ...

    As 11:00 approaches, an expedition is formed to locate a new can to live. A returning bacterium announces, "Hooray, we have located 3 more empty cans. Thats three times more space than we've ever had in our entire existence.
    --B--How much more time has the expedition provided?
    At 11:00 the can is filled completely minus 1 bacterium.

    If the amount of bacteria is measured in cans the number of cans can be calculated by:

    c(t) = 2^t

    There are 3 cans available:

    3 = 2^t~\iff~ \ln(3)=t \cdot \ln(2)~\iff~ t =\frac{\ln(3)}{\ln(2)} ~ \approx ~1.5849625\ min = 1 min 35 s
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Mar 2008
    Posts
    148
    Ok so the can is able to hold (8)(2^{57}) = 2^{60} bacteria since at minute n there are 2^n bacteria and after 57 minutes the can is 1/8 full. Provided that a dead bacteria disintigrates into the ether upon death and doesn't take up any room in the can then at 11:00 there will be 2^{60} - 1 bacteria in the can, (the first guy died). In the next minute however the population will double and only 1 more bacterium dies (the 1 created at 10:01). Since 2(2^{60} -1) - 1 >> 2^{60} the party will end at 11:01 (4 minutes after the first said "Party").

    Given 3 times the space, the group is able to support (3)2^{60} bacteria. 2(2^{60} -1) - 1 < (3)2^{60} < 2(2(2^{60} -1) - 1) - 2
    -the least value in this inequality is the population at 11:01, the greatest value is the population at 11:02, so it bought them 1 minute.

    What do you think? I guess I am assuming the doubling happens instantly at the end of the minute and immediately after the bacteria 1 hour old die. (more of a mind game interpretation than a population growth).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: June 9th 2011, 09:54 AM
  2. log and exponential functions
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: May 3rd 2009, 03:27 PM
  3. Exponential Functions
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: October 16th 2008, 09:28 PM
  4. Exponential Functions...
    Posted in the Pre-Calculus Forum
    Replies: 8
    Last Post: August 26th 2008, 09:51 AM
  5. Exponential Functions
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: May 12th 2008, 08:50 PM

Search Tags


/mathhelpforum @mathhelpforum