College algebra

Question

The number of bacteria in a culture is increasing according to the law of exponential growth. After 2 hours, there are 100 bacteria, and after 5 hours, there are 800 bacteria. How many bacteria will there be after 6 hours?

Answer 1

2 hours 100

5 hours 800

6 hours ?

Just an initial rough guess might be about 1000

A = Pe^rt where A = amount at time t hours, r = percentage rate of increase, P = amount at time t=0

100 =Pe^2r

100/P =e^2r

ln100- lnP = 2r

lnP = ln100 -2r

800 =Pe^5r

ln800/P = e^5r

ln800 -5r= lnP

ln100-2r= lnP

subtract

ln800 -ln100 -5r--2r = lnP-lnP

3r = ln800-ln100

r = (ln800-ln100)/3

= (ln(800/100)/3

= ln8/3

= about .69315

r= about 69.315%

lnP = ln100 -2r = 4.6052 - 1.3863 = 3.2189

P = e^3.2189 = about 24.9999. = amount at time t=0

P(0) = 25e^.695(0)=25(1) = 25

A(2) = 25e^.63(2) = 25e^1.38962 = 25(4) = 100

A(6) = 25e^.63(6) = 25e^3.78 = 25(43.8)

bacteria amount = about 1098 in 6 years