Question

A population of protozoa develops with a constant relative growth rate of 0.4964 per member per day. On day zero the population consists of five members. Find the population size after eight days. Part 1 of 3 Since the relative growth rate is 0.4964, then the differential equation that models this growth is dP dt = x.4964 .

Answer 1

Answer:

The population size after eight days is about 265

Step-by-step explanation:

This is an example of an exponential growth model. A quantity y that grows or decays at a rate proportional to its size fits in an equation of the form

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

where k is a positive constant. Its solutions have the form

$y=y_{0}e^{kt}$,

where $y_{0} =y(0)$ is the initial value of y.

The population size can be calculated by using the below formula:

$P(t)=P(0)e^{kt}$ where $P(0)$ is the population on day zero.

Let t be the time in days,

We know from the information given that:

• k = 0.4964 per member per day and

• The day zero (t = 0) the population size is 5 (P(0) = 5)

To find the population size after eight days

Substitute P(0) = 5, k=0.4964 in $P(t)=P(0)e^{kt}$

Then

$P(t)=5e^{0.4964\cdot t}$

Now we calculate P(t) when t = 8 days

$P(8) = 5e^{0.4964\cdot 8}\\P(8) = 5e^{3.9712}\\e^{3.9712}=53.04815\dots \\P(8) = 5 \cdot 53.04815\dots\\P(8) = 265.24075\dots \approx 265$

Therefore the population size after eight days is about 265