Question

The population \(p\) of a small town after \(x\) years can be modeled by the function \(p=6850(1.03)^x\). What is the average rate of change in the population over the first 6 years? Justify your answer.

Short answer

To get the short answer, plug in the calculated or given values of \(p_6\) and \(p_0\) into the average rate of change formula \((p_6 - p_0) / 6\) and solve.

Step-by-step solution

Identify the function

The population \(p\) of the town after \(x\) years is given by the function \(p=6850(1.03)^x\). This is an exponential growth function, where 6850 is the initial population, 1.03 is the growth rate, and \(x\) is the time in years.

Calculate population at year 0

Substitute \(x = 0\) into the equation to get the initial population \(p_0\), so \(p_0 = 6850(1.03)^0 = 6850\).

Calculate population at year 6

Substitute \(x = 6\) into the equation to get the population at year 6, \(p_6 = 6850(1.03)^6\). Solve to get the numerical value of \(p_6\).

Compute the average rate of change

The average rate of change over the first 6 years is given by the equation \((p_6 - p_0) / 6\). Substitute the calculated values of \(p_6\) and \(p_0\) into the equation and solve to get the average rate of change.

Key concepts

Exponential Growth Function

Imagine a small town with a few hundred residents that begins to grow rapidly due to a new industry bringing jobs and attracting new residents. This kind of rapid increase can often be described by what mathematicians call an exponential growth function. The formula given in our problem, p = 6850(1.03)^x, is a classic example of such a function. Here, 6850 represents the initial population of the town. The number 1.03 is the growth rate, which means every year, the population increases by 3% compared to the previous year.

The power of exponentials lies in compounding, where increases are not just additive; they're multiplicative. So, in year one, the population is 3% more than 6850, but in year two, it's 3% more than the population at the end of year one, and so on. This results in a steep increase that can dramatically change the town's demographic landscape over time. Exponential functions like this are essential for modeling real-world phenomena where changes occur at rates proportional to the current value, such as interest in banking, radioactive decay in physics, or indeed, population growth in ecology.

Function Modeling

When we talk about function modeling, we are referring to the use of mathematical functions to represent real-world scenarios. A function, in its simplest definition, is a relation that uniquely associates members of one set with members of another set. In our case, the sets are 'years since the start' and 'town population'.

By plugging in different values for x (the number of years), the exponential function gives us the corresponding population p. This process of substituting values into the function's formula to predict outcomes is a powerful tool, especially when planning for future resource needs or understanding how certain factors will affect growth over time. Moreover, function modeling is not only limited to growth; it can also describe reduction, oscillation, or other complex behaviors seen in different contexts.

Population Growth

When it comes to population growth, the relationship between time and population is not always straightforward. While a linear growth model assumes the same number of individuals are added to the population each year, many species, including humans, tend to grow in numbers at a rate that depends on the current population. This is where our exponential growth model comes into play.

In this scenario, the small town starts with a population of 6850 and then grows by a rate of 3% per year. If you want to analyze the change between two particular moments in time, you can compute the average rate of change. Remember that this is the slope of the line connecting two distinct points on our function's graph. From year 0 to year 6, this tells you how swiftly the population has increased on average each year. Knowing this can help town planners and policy-makers make informed decisions regarding required services, infrastructure expansion, and sustainability efforts.