# A sample of 2100 bacteria selected from this population reached the size of 2169 bacteria in one and a half hours. How do you find the hourly growth rate parameter?

May 12, 2017

Hourly growth parameter is 2.155% per hour.

#### Explanation:

An hourly growth parameter of $h$, indicates that in an hour $x$ number of bacteria become $x {e}^{h}$ and in $n$ hours it would be $x {e}^{n h}$

As such in one and a half hour, $2100$ becomes $2100 \times {e}^{1.5 h}$

Hence $2100 \times {e}^{1.5 h} = 2169$

i.e. ${e}^{1.5 h} = \frac{2169}{2100} = 1.023857$

and $1.5 h = \ln 1.023857 = 0.0323287488$

and $h = 0.0323287488 \times \frac{2}{3} = 0.02155$ or say 2.155%

Another way which is also used could be as given below.

An hourly growth parameter of $h$, indicates that in an hour $x$ number of bacteria become $x \left(1 + \frac{h}{100}\right)$

and in $n$c hours they become $x {\left(1 + \frac{h}{100}\right)}^{n}$

as in sample $2100$ bacteria reach the size of $2169$ in one and a half hour, we have

$2169 = 2100 \left(1 + \frac{h}{100}\right) \left(1 + \frac{h}{200}\right)$

or $\left(1 + \frac{h}{100}\right) \left(1 + \frac{h}{200}\right) = \frac{2169}{2100} = \frac{723}{700}$

This leads to $h = 2.179$ - (up to $3$ decimal places using Goal seek in MSExcel)

Hence, hourly growth parameter is 2.179% per hour.