Find the population after t hours.?

A cell of some bacteria divides into two cells every 10 minutes.The initial population is 4 bacteria. Hint: The calculation will be easier if you let the unit of time be 10 minutes, so that one hour is (60/10) units of time.


a. Find the population after t hours.


b. Find the population after 4 hours.


c. Find the population after 16 hours.





2.A bacteria culture starts with 700 bacteria and grows at a rate propotional to its size.After 2 hours there are 1400 bacteria.


a. Find the population after t hours


b. Find the population after 5 hours.


c. When will the population reach 1850 ?

Find the population after t hours.?
1) P = 4*2^t, where t=1 means 10 minutes, t=2 means 20 min.





Check: at t=0, there were 4. After 10 minutes, there will be 4*2 = 8, after 20 minutes there will be 4*2^2 = 16.





a) pop after T hours means 6 units of time*T or


P = 4*2^(6T), T in hours.





b) Pop after 4 hours means


P = 4*2^(24).





Do the rest for this problem.





2.A bacteria culture starts with 700 bacteria and grows at a rate propotional to its size.After 2 hours there are 1400 bacteria.





dP/dt = aP, separate


dP/P = a dt, integrate


ln(P) = at + C, so


e^(at+C) = P or


Ce^(at) = P. At t=0, there were 700, so C = 700, or





700 e^(at) = P. at t = 2 hours, we have 1400, so


700 e^(2a) = 1400, div by 700 or


e^(2a) = 2, take ln of both sides, or


2a = ln(2), so


a = ln(2)/2. So formula is








700 e^(t*ln(2)/2) = P.





Do the rest.