Question

Suppose the number of bacteria in a certain population increases according to a continuous exponential growth model. a sample of 2200 bacteria selected from the population reached the size 2300 bacteria in one and a half hours. Find the hourly growth rate parameter.

Answer 1

Answer:

The hourly growth rate parameter of the bacteria is 3%.

Step-by-step explanation:

Continuous exponential growth formula:

$A=Pe^{rt}$

A= Population of bacteria after t hours

P = Initial population

r= growth rate

t= time in hours

Given that,

A sample of 2200 bacteria selected from the population reached the size 2300 bacteria in one and half hours.

A=2300, P=2200, t $=1\frac12$ hours =$\frac32$ hours,r=?

$\therefore2300=2200e^{\frac32r}$

$\Rightarrow2200e^{\frac32r}=2300$

$\Rightarrow e^{\frac32r}=\frac{2300}{2200}$

$\Rightarrow e^{\frac32t}=\frac{23}{22}$

Taking ln both sides

$\Rightarrow ln( e^{\frac32r})=ln(\frac{23}{22})$

$\Rightarrow {\frac32r}=ln(\frac{23}{22})$

$\Rightarrow r}=\frac{ln|\frac{23}{22}|}{\frac32}$

$\Rightarrow r}\approx0.030$

$\Rightarrow r}=3\%$

The hourly growth rate parameter of the bacteria is 3%.