Question

The doubling time of a bacterial population is 15 minutes. After 80 minutes, the bacterial population was 60000.

-What was the initial population of bacteria?

-Using your rounded answer from above, find the size of the bacterial population after 3 hours.

Answer 1

Answer:

The initial population of bacteria was 1488.

The size of the bacterial population after 3 hours will be 6,084,093

Step-by-step explanation:

The population of bacteria after t minutes can be modeled by the following equation.

$P(t) = P(0)e^{rt}$

In which P(0) is the initial population and r is the growth rate.

The doubling time of a bacterial population is 15 minutes.

This means that $P(15) = 2P(0)$. We use this to find r.

$P(t) = P(0)e^{rt}$

$2P(0) = P(0)e^{15r}$

$e^{15r} = 2$

$\ln{e^{15r}} = \ln{2}$

$15r = \ln{2}$

$r = \frac{\ln{2}}{15}$

$r = 0.0462$

After 80 minutes, the bacterial population was 60000.

This means that $P(80) = 60000$. So

$P(t) = P(0)e^{0.0462t}$

$60000 = P(0)e^{0.0462*80}$

$P(0) = \frac{60000}{e^{0.0462*80}}$

$P(0) = 1488$

This means that the initial population of bacteria was 1488.

Using your rounded answer from above, find the size of the bacterial population after 3 hours.

t is in minutes, so this is P(3*60) = P(180).

$P(t) = 1488e^{0.0462t}$

$P(180) = 1488e^{0.0462*180}$

$P(180) = 1488e^{0.0462*180}$

$P(180) = 6,084,093$

The size of the bacterial population after 3 hours will be 6,084,093