Bacteria culture?

A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 280.


(a) Find an expression for the number of bacteria after t hours.


P(t) = cells





(b) Find the number of bacteria after 2 hours.


P(2) = cells





(c) Find the rate of growth after 2 hours.


P'(2) = cells/hour





(d) When will the population reach 10,000?


t = hours

Bacteria culture?
The population P(t) of something that grows at a rate proportional to its size can be represented by the exponential function





P(t) = P(0)e^(kt)





Since we know P(0) = 100 and P(1)=280 we can substitute to find the growth rate factor k





280 = 100e^(k*1) so





2.8 = e^k and taking ln both sides





ln2.8 = klne but lne = 1 so





1.0296 = k





(a) So our growth equation is P(t) = 100e^(1.0296t)





(b) After two hours we have P(2)=100e^(1.0296*2) = 784





(c) Differentiating (a) P'(t) = 100e^(1.0296t) * 1.0296 and





P'(2) = 100e^(1.0296*2)*1.0296





= 288 cells/hr





(d) Finally when P(t) = 10,000





10,000 = 100e^1.0296t





100 = e^1.0296t and taking ln both sides





ln100 = 1.0296t





t = ln100/1.0296 = 4.47 hours