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Differential equations play a crucial role in modeling various real-world phenomena, and population growth is one such area. When we talk about differential equations, we're referring to equations that involve functions and their derivatives. These equations model how a particular quantity changes over time.
The growth of a population can often be represented as a rate of change, which is where differential equations come in handy. In the context of our exercise, the given growth rate function, \( R(t) = 1.5 + 0.3\sqrt{t} + 0.006 t^2 \), is in differential equation form. The function \( R(t) \) tells us the rate at which the population is increasing every year, expressed in thousands of people.
Solving a differential equation allows us to predict future values of a quantity—in this case, the future population. By integrating the growth rate function, the differential equation helps us determine the total change in population over a specified period.

The definite integral is a powerful tool in calculus, used to compute the total accumulation of a quantity over a certain interval. In our exercise, the definite integral helps determine the total population increase from 1985 to 1994.
By integrating the growth rate function \( R(t) = 1.5 + 0.3\sqrt{t} + 0.006 t^2 \) over the interval from \( t = 0 \) to \( t = 9 \), we can calculate the total number of thousands of people added to the city's population during these 9 years. Mathematically, this is expressed as finding \[ \int_{0}^{9} (1.5 + 0.3\sqrt{t} + 0.006 t^2) \, dt \].
The process involves calculating the integral of each component of \( R(t) \), which when evaluated from \( t = 0 \) to \( t = 9 \), gives the total change in the population. This method effectively accumulates the yearly changes in population over the given time period.

Understanding the growth rate function is key to modeling population dynamics. This function characterizes how the population changes over time and is essential for making predictions about future populations.
The given function, \( R(t) = 1.5 + 0.3\sqrt{t} + 0.006 t^2 \), provides the rate of population increase "per year" and is described in thousands of people. It incorporates both constant and variable components:

• A constant term (\(1.5\)), which contributes a steady increase annually.
• A square root term (\(0.3\sqrt{t}\)), which implies a diminishing growth effect as time passes, suggesting early acceleration.
• A quadratic term (\(0.006 t^2\)), indicating increasing growth impact over time, representing long-term acceleration.

Analyzing such a growth rate function allows us to understand how different factors contribute to population growth and how these factors combine to influence the total growth trajectory over time.