bacteria word problem

Question

A bacteria culture starts with 560 bacteria and grows at a rate proportional to its size. After 3 hours there will be 1680 bacteria.

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

(b) What will be the population of bacteria be after 8 hours?

(c) How long will it take for the population to reach 1570? Give your answer accurate to at least 2 decimal places.

Answer 1

This is a problem in exponential growth.  P(t) = 560e^kt

P(3) = 560e^3k = 1680

Divide by 560: e^3k = 3

Take natural logs: 3k = ln 3

k = (ln 3)/3 = 0.3662040962

 

Now you can answer (a) and (b) easily.

 

(c)  P(t) = 560e^kt = 1570

e^kt = 2.803571429

kt = ln(2.803571429)

t = 1.030894...)/k = 2.815 hours

Answer 2

(a)

 

dP/dt = kP

 

dP/P = kdt

 

ln(P) = kt + ln(c), c = unknown constant

 

P = e^kt * c

 

P(0) = c

 

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

 

At t = 3 hrs,

 

1680 = 560 e^(3k)

 

k = ln(1680/560) / 3 = 0.3662 hours^-1

 

P(t) = 560 e^(0.3662 t), t in hours

 

(b)

 

P(8) = 560 e^(0.3662*8) = 10483

 

(c)

 

1570 = 560 e^(0.3662 t)

 

t = ln(1570/560) / 0.3662 = 2.815 hrs