A population of protozoa develops with a constant relative growth rate of 0.5589 per member per day. On day zero the population consists of five members. Find the population size after four days. Round to the nearest whole number.
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A population of protozoa develops with a constant relative growth rate
of 0.5589 per member per day.
On day zero, the population consists of five members.
Find the population size after four days. Round to the nearest whole number.
We are told: .$\displaystyle \frac{dP}{dt}\;=\;0.558P$
Then we have: .$\displaystyle \frac{dP}{P} \:=\:0.588\,dt$
Integrate: .$\displaystyle \ln P \:=\:0.558t + c\quad\Rightarrow\quad P \;=\;e^{0.558t+ c}$
. . and we have: .$\displaystyle P \;=\;e^{0.558t}\cdot e^c\quad\Rightarrow\quad P \;=\;Ce^{0.558t}$
We are told that when $\displaystyle t = 0,\;P = 5$
. . So we have: .$\displaystyle 5 \;=\;Ce^0\quad\Rightarrow\quad C \:=\:5$
The population function is: .$\displaystyle P(t) \;=\;5e^{0.558t}$
In four days: .$\displaystyle P(4) \;=\;5e^{(0.558)(4)} \;=\;46.59242212\;\approx\;\boxed{47}$