91Ó°ÊÓ

Understanding the rate of population growth is fundamental in demographics and environmental studies.
In our case, the rate at which the population of a rural town grows is expressed mathematically by the function
\(P^{\text{prime}}(t)=30(1+\sqrt{t})\)
This represents the derivative of the population with respect to time, capturing the change in population size over a given period.

In the context of our exercise, this function tells us how many people are added to the town's population each year, starting from the initial record-keeping point, \(t=0\).
One important observation about this rate function is its dependence on \(t\), indicating that the population growth rate increases with time.

To find out how much the population has grown after a certain number of years, \(t\), we need to integrate this rate of growth over the time interval in question.
By doing so, we accumulate the growth rates over each infinitesimal moment within that interval to get the total population change.

Integration is one of the two main operations in calculus, with differentiation being the other.
It is the process of finding the area under a curve defined by a function, representing the accumulation of quantities.

In our exercise, we utilize integration to find the total change in population over a certain time period.
The given rate of growth function, \(P^{\text{prime}}(t)=30(1+\sqrt{t})\), needs to be integrated with respect to time to find the general population function \(P(t)\).

Using integration techniques, we split the given function into simpler parts that we can integrate individually.
We then apply the power rule: \(\int t^n dt = \frac{t^{n+1}}{n+1}+C\)
to the term \(t^{\frac{1}{2}}\) in our function and add a constant of integration, \(C\), which will be determined using an initial condition.

An initial value problem in calculus is a type of differential equation along with a specified value, called the initial condition, which the solution must satisfy at a given point in its domain.

In the context of our population growth example, the initial value is the population size when record-keeping began: \(P(0)=250\).
This crucial piece of information allows us to find the constant of integration, \(C\), after integrating the rate of growth function to obtain the general solution for the population, \(P(t)\).

The process involves substituting the initial condition back into the integrated function.
From there, we solve for \(C\), leading us to the specific equation that models the population growth: \(P(t) = 30t + 20t^{\frac{3}{2}} + 250\).
With this formula, we can predict the population at any future time \(t\), thus solving the initial value problem presented in the exercise.