Question

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?

Answer 1

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 `xe^h` and in `n` hours it would be `xe^(nh)`

As such in one and a half hour, `2100` becomes `2100xxe^(1.5h)`

Hence `2100xxe^(1.5h)=2169`

i.e. `e^(1.5h)=2169/2100=1.023857`

and `1.5h=ln1.023857=0.0323287488`

and `h=0.0323287488xx2/3=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(1+h/100)`

and in `n`c hours they become `x(1+h/100)^n`

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

`2169=2100(1+h/100)(1+h/200)`

or `(1+h/100)(1+h/200)=2169/2100=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.