Find the initial population?

A bacteria culture grows with constant relative growth rate. After 2 hours there are 600 bacteria and after 8 hours the count is 75,000.





(a) Find the initial population.


(b) Find an expression for the population after t hours.


(c) Find the number of cells after 5 hours.


(d) Find the rate of growth after 5 hours.


(e) When will the population reach 200,000?

Find the initial population?
dy/dt = ky.





y = Ae^(kt)





If t = 2, y = 600


t = 8, y =75000


solve for A and e^k here.





600 = Ae^(2k)


75000 = Ae^(8k)


to eliminate A, divide the 2nd by the first. Then you will get your e^k. Then you can find A.





125= e^(6k) then e^k = 125^(1/6) = 5^(3/6) = √5.


600 = A*5.





To answer:


(a) the answer is simply A.


(b) the answer is the whole equation with A %26amp; e^k determined. Note: e^(kt) = (e^k)^t.


(c) when t = 5 find y.


(d) dy/dt = k(y5) , y5 is the answer in (c).


(e) when y = 200,000 find t.





You probably can do most of this now.








d: