A population of protozoa develops with a constant relative growth rate of 0.8106 per member per...

Question:

A population of protozoa develops with a constant relative growth rate of 0.8106 per member per day. On day zero, the population consists of 8 members.

Find the population size after 7 days.

Exponential Growth

Exponential growth function of the form {eq}y = A_0 e^{kt} {/eq} where {eq}A_0 {/eq} is the initial value of the function or the value of y at {eq}t = 0 {/eq} and {eq}k {/eq} is the growth rate constant. An exponential growth function can be used to model the growth of microorganisms where {eq}A_0 {/eq} is the initial number of individuals. The exponential growth function is the general solution to the differential equation {eq}\displaystyle \frac{dy}{dt} = ky {/eq}.

Suppose a protozoan population grows at a constant rate of 0.8106 member per day, then we can express this in terms of the differential equation

{eq}\displaystyle \frac{dP}{dt} = 0.8106 P {/eq}

where P is the number of protozoans at any time, t.

This differential equation has the general solution,

{eq}P(t) = P_0 e^{0.8106 t} {/eq}

where {eq}P_0 = 8 {/eq} is the number of protozoans at day zero.

Now to solve for the number of protozoans after 7 days, we plug {eq}t = 7 {/eq} into {eq}P(t) {/eq}.

{eq}\displaystyle \begin{align*} P(t) &= P_0 e^{0.8106 t} \\ P(7)&= 8 e^{0.8106(7)} \\ &= \boxed{ 2330 \text{ protozoans}} \end{align*} {/eq}

Therefore, after 7 days there are {eq}\boxed{ 2330 \text{ protozoans}} {/eq}.