Jenna T.

asked • 04/15/17

How much bacteria will there be at midnight?

During a research experiment it was found that the number of bacteria grew at a rate proportional to its size. At 6 am there were 6000 bacteria present in the culture at noon there were 6,600 bacteria present. How many bacteria will there be at midnight?

Mark M.

Growth proportional to size requires differential equations. Are you at that level?
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04/15/17

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Arturo O. answered • 04/15/17

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Exponential growth is given by
 
A(t) = A0ekt
 
Note:  This expression comes from solving the differential equation for growth rate proportional to size, but I assume you are allowed to use it in your class.
 
A(t) = amount at time t (use hours for the time unit)
A0 = initial amount
Get k from the given information
 
t = 6 hours (from 6:00 am to noon)
A(6) = 6600
A0 = 6000
 
6600 = 6000ek(6)
 
k = ln(6600/6000) / 6 = 0.015885 hrs-1
 
A(t) = 6000e0.015885t, t in hours as of 6:00 am
 
At midnight, t = 18 hours
 
A(18) = 6000e0.015885(18) ≅ 7986
 
 
 
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Arturo O.

By the way, if you want to see a derivation of A(t) from differential equations, we can can get into that.  Basic calculus is sufficient.  Just post a comment requesting it.
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04/15/17

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