Question

A population of protozoa develops with a constant relative growth rate of 0.6671 per member per day. On day zero the population consists of 4 members. Find the population size after 7 days. Since the relative growth rate is 0.6671, then the differential equation that models this growth is

Answer 1

The population size after 7 days is about 427.

Explanation:

If
`y(t)` is the value of a quantity
`y` at time
`t` and the if the rate of change of
`y` with respect to
`t` is proportional to its size
`y(t)` at any time, then

`(dy)/(dt) =ky`

where
`k` is a constant.

This equation is sometimes called the law of natural growth (if
`k>0`).

The only solutions of the differential equation
`(dy)/(dt) =ky` are the exponential functions

`y(t)=y(0)e^(kt)`

Let
`P` be the population size and let
`t` be the time variable, measured in hours. Since the relative growth rate is 0.6671, then the differential equation that models this growth is

`(dP)/(dt) =0.6671\cdot P`

According with the above information the solution to this differential equation is

`P(t)=P(0)e^(0.6671t)`

On day zero the population consists of 4 members
`P(0)=4`.

Therefore, the population size after 7 days is

`P(7)=4e^(0.6671\cdot 7)\\\\P(7)=4e^(4.6697)=4\cdot \:106.66573=426.66295`

about 427.