bacteria popluation growth

Question

Example. A bacteria population experiences exponential growth with a constant growth rate of r. After 4 hours, the population increases by 30%. When will the population double?

Solution: We will use A(t)=Pe^rt.

after 4 hours P = P +.3P = 1.3P

so setting t - 4 we have

1.3p = pe^4r

1.3=e^4r

ln(1.3) = ln(e^4r) =4r

so r = ln(1.3) /4

returning to formula

A(t) =pe^ln(1.3)/4t

=p(1.3)t/4

2p = p(1.3)^t/4

ln(2)= ln(1.3^t/4) = t/4 * ln(1.3)

t = 4*ln(2) / ln(1.3)

question

A bacteria population grows according to A(t) = Pe^rt

where t is measured in hours. If the growth rate is 7%, after how many hours will the population double? (Give your answer to the nearest hundredth.)

Answer 1

after 4 hours P increases by 30% when does it double? when it increases by 100%

1.3p =pe^4r

1.3 = e^4r

ln1.3 = 4r

r= ln1.3/4

2= e^rt= e^.25tln1.3

ln2 =.25tln1.3

t = 4ln2/ln1.3= about 10.57 hours

r=.07

2 =e^.07t

t = ln2/.07= 9.90 hours to double

Answer 2

A(t) = 2P. Which means that the population is doubled at time t.

Now you can write 2P = Pe^rt

The P's cancel. You know r. Now you can solve for t.

Give it a try! Post your answer here and we can check it for you.