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A bacteria culture initially contains 2500 bacteria and doubles every half hour. The formula for the population is p(t)=2500e^(kt) for some constant k.

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1 Answer

To answer this questions we must first find the constant k. If we find k then we can calculate our population for anytime.
 
I am going to assume that time t is measured in hours as the problem statement doesn't specify.
 
The population doubles every half hour, therefore:
 
p=5000 when t=0.5 hours
 
Plug these values in the equation to obtain:
 
5000=2500e^(0.5k)
 
Simplify...
 
2=e^(0.5k)
 
Square both sides
 
4=e^k
 
k=ln(4)
 
Now we have k so we can plug it in the original equation and obtain p for any time
 
p(t)=2500e^(ln(4)t), which simplifies to
 
p(t)=2500(4)^t
 
Now evaluate for t=100 minutes (1.67 hours) and t=4 hours
 
p(1.67)=2500(4)^1.67
 
p(1.67)=25315
 
p(4)=2500(4)^4
 
p(4)=640000