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

Answer:

The population size after 7 days is about 427.

Step-by-step 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

                                                  $\frac{dy}{dt} =ky$

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 $\frac{dy}{dt} =ky$ 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

                                                 $\frac{dP}{dt} =0.6671\cdot P$

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.