A bacteria culture starts with 580 bacteria and grows at a rate proportional to its size. After 3 hours there will be 1740 bacteria.

Question

A bacteria culture starts with 580 bacteria and grows at a rate proportional to its size. After 3 hours there will be 1740 bacteria.

 

 

(a) Express the population after t hours as a function of t.
population:

(b) What will be the population after 7 hours?


(c) How long will it take for the population to reach 2870 ?

Answer 1

(a)

 

This is an exponential growth problem.  The population as a function of time obeys the equation

 

P(t) = P₀e^kt

 

where P₀ is the population at time t=0.  We need to find k.  At t=3 hours,

 

1740 = 580e^k(3)

 

k = (1/3) ln(1740/580) hours^-1 ≅ 0.3662 hours^-1

 

P(t) = 580e^0.3662t

 

(b)

 

P(7) = 580e^0.3662(7) ≅ 7528

 

(c)

 

2870 = 580e^0.3662t

 

Solve for t.

 

t = ln(2870/580) / 0.3662 hours ≅ 4.367 hours