MarkFL said:We know the population will take the form:
\(\displaystyle P(t)=P_0e^{kt}\)
We are then given:
\(\displaystyle P(3)-P(2)=2455\)
\(\displaystyle P(5)-P(4)=4314\)
This will give you 2 equations and 2 unknowns...can you proceed to find the initial population $P_0$?
rayne said:What would be the k value?
MarkFL said:This would also have to be algebraically determined. Let's look at the first equation I gave:
\(\displaystyle P(3)-P(2)=2455\)
This then becomes (using the definition of $P(t)$):
\(\displaystyle P_0e^{3k}-P_0e^{2k}=2455\)
Now, if we factor on the left, we obtain:
\(\displaystyle P_0e^{2k}\left(e^k-1\right)=2455\)
And solving for $P_0$ we then get:
\(\displaystyle P_0=\frac{2455}{e^{2k}\left(e^k-1\right)}\)
Now, if we state the second equation I gave, we have (after factoring):
\(\displaystyle P_0e^{4k}\left(e^k-1\right)=4314\)
At this point we may substitute for $P_0$ that we obtained above:
\(\displaystyle \frac{2455}{e^{2k}\left(e^k-1\right)}e^{4k}\left(e^k-1\right)=4314\)
Now you may simplify, then solve for $e^k$, and then you can determine $P_0$.
Finding the initial population is important because it serves as the starting point for studying a population and understanding its characteristics and trends over time.
The initial population can be determined through various methods, such as census data, historical records, or scientific surveys. It is important to use reliable and accurate data sources for obtaining the initial population.
Several factors can affect the initial population, including birth and death rates, immigration and emigration, natural disasters, and human activities such as hunting or deforestation. It is important to consider these factors when determining the initial population.
Estimating the initial population accurately is crucial for making informed decisions and predictions about the population's future growth and sustainability. It also helps in understanding the impact of various factors on the population and implementing effective management strategies.
Yes, the initial population can change over time due to various factors such as population growth, environmental changes, and human activities. It is important to regularly update and re-evaluate the initial population to accurately track and manage the population's changes.