document.write( "Question 121172: There are initially 3 E-coli bacteria present in a hamburger. If the doubling time for E-coli is 5 minutes, how many E-coli bacteria will be present after 20 minutes? \n" ); document.write( "
Algebra.Com's Answer #88922 by bucky(2189)\"\" \"About 
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Think of it this way ... each 5 minutes the number of E-coli doubles from what you had at the
\n" ); document.write( "start of the 5 minute period.
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\n" ); document.write( "You start with 3 bacteria and after 5 minutes you have double that ... or 6 bacteria.
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\n" ); document.write( "In the next 5 minute period you start with 6 and you double that, so you end that 5 minute period
\n" ); document.write( "with 12 bacteria.
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\n" ); document.write( "In the third 5 minute period you start with 12 and you double that so you finish that period
\n" ); document.write( "with 24 bacteria.
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\n" ); document.write( "And in the last 5 minute period you start with 24 and end up the period with 48 bacteria.
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\n" ); document.write( "So in 20 minutes (four 5-minute periods) you go from 3 bacteria to 48 bacteria.
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\n" ); document.write( "Can you find a formula that will cover that process so that you won't have to do the same
\n" ); document.write( "sort of process for longer periods of time? Sure we can. All we have to do is to find the
\n" ); document.write( "number of 5-minute periods involved and then multiply the starting value by 2 for each
\n" ); document.write( "5-minute period. So if we call the number we start with E, and we have six 5-minute periods,
\n" ); document.write( "we just multiply E by 2 six times:
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\n" ); document.write( "E*2*2*2*2*2*2
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\n" ); document.write( "but you know that 2*2*2*2*2*2 equals 2^6 so you can shorten the formula to:
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\n" ); document.write( "Number of bacteria = E*(2^6)
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\n" ); document.write( "and you can make it more general by recognizing that the exponent is the total time in minutes
\n" ); document.write( "divided by 5 minutes or T/5. This makes the equation:
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\n" ); document.write( "Number of bacteria = E*(2^(T/5))
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\n" ); document.write( "Let's try the original problem by letting E = 3 bacteria and T = 20 minutes. Our equation becomes:
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\n" ); document.write( "Number of bacteria = 3*(2^(20/5)) = 3*2^4 = 3*16 = 48 bacteria.
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\n" ); document.write( "This is the answer we got so our formula works OK.
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\n" ); document.write( "Now let's change the problem. Start again with E = 3 bacteria and find out how many bacteria
\n" ); document.write( "there are in an hour (60 minutes):
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\n" ); document.write( "Number of bacteria = 3*(2^(60/5)) = 3*2^12 = 3*4096 = 12,288 bacteria.
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\n" ); document.write( "So if you start with 3 bacteria and double every 5 minutes for an hour, you end up with a
\n" ); document.write( "colony of 12,288 bacteria.
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\n" ); document.write( "Hope this helps you to understand the problem a little better.
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