Problem 70 - Contemporary Precalculus

Question

The U.S. Department of Commerce estimated that there were 54 million Internet users in the United States in 1999 and 85 million in 2002 . (a) Find an exponential function that models the number of Internet users in year \(x,\) with \(x=0\) corresponding to 1999 (b) For how long is this model likely to remain accurate? [Hint: The current U.S. population is about 305 million.]

Step-by-Step Solution

Answer

Answer: The exponential function that models the number of internet users in the United States is given by y = 54 * (cubic root (85/54))^x. This model is likely to remain accurate as long as the number of internet users is considerably less than the total population of the United States.

Step 1: Determine the exponential function form

The general form of an exponential function is given by: \[y = a b^x\] where \(y\) represents the number of internet users, \(x\) represents the time in years since 1999, and \(a\) and \(b\) are constants we need to determine using the given data points.

Step 2: Substitute the data points into the exponential function

We can plug in the given data points, (0, 54) and (3, 85), into the exponential function to create two equations with \(a\) and \(b\): 1. For the data point (0, 54): \[54 = a \cdot b^0\] 2. For the data point (3, 85): \[85 = a \cdot b^3\]

Step 3: Solve for the constants 'a' and 'b'

From the first equation, we have: \[54 = a \cdot b^0 \Rightarrow 54 = a \cdot 1 \Rightarrow a = 54\] Now, substitute this value of \(a\) into the second equation: \[85 = 54 \cdot b^3\] Then, solve for \(b\): \[b^3 = \frac{85}{54}\] \[b = \sqrt[3]{\frac{85}{54}}\]

Step 4: Form the exponential function

Now that we have the values of \(a\) and \(b\), we can write the exponential function that models the number of Internet users: \[y = 54 \cdot \left(\sqrt[3]{\frac{85}{54}}\right)^x\]

Step 5: Discuss the model's accuracy

The exponential function derived in the previous step represents the number of Internet users as a function of time, where \(x=0\) corresponds to the year 1999. This model assumes that the growth of internet users will continue at an exponential rate. However, as the population of the United States is about 305 million, it is reasonable to assume that the model's accuracy will decrease once the number of internet users approaches or exceeds the total population. In other words, the model is most accurate for the years in which the number of Internet users in the United States is considerably less than the total population.

Key Concepts

Internet Users

In the late 1990s and early 2000s, the expansion of the internet led to a significant increase in the number of internet users. This phenomenon can be mathematically explained using an exponential growth model. Exponential growth occurs when the rate of increase is proportional to the current size, leading to faster and faster growth over time. With an initial count of 54 million internet users in 1999, the United States saw this number rise to 85 million by 2002.

When measuring internet user growth, it's essential to understand that the exponential function allows us to predict future numbers based on past data. As calculated, the exponential formula: \(y = 54 \cdot \left(\sqrt[3]{\frac{85}{54}}\right)^x\)models the increase from 1999 onwards. In this formula, \(y\) represents the number of users, while \(x\) represents years since 1999.

This kind of model assumes a continuous growth rate, ideal for predicting trends over short periods while acknowledging limitations when the number of users nears a ceiling, such as total population size.

Population Growth

Population growth is another classic example of exponential growth. In contrast to internet users, which can rise rapidly due to technological adoption, human populations grow based on birth rates, death rates, and migration. This kind of growth tends to be slower but can still be modeled using exponential functions under the right conditions.

In the context of the exercise, when we compare internet user growth to population size, an important aspect is highlighted. If the growth rate turns out to be unsustainable when the number of users potentially exceeds the total population of a location, such as the United States with 305 million people in this context.

Generally, rapid increases in populations can strain resources and spark discussions on sustainability. Nonetheless, mathematical modeling offers a helpful tool for demography and urban planning, allowing for insights into future challenges and demands based on current trends.

Mathematical Modeling

Mathematical modeling is a powerful method used to describe real-world phenomena systematically. It transforms complex systems into mathematical formulas, like the one used for predicting the number of internet users over time. Models such as exponential functions provide clarity by simplifying assumptions and giving estimates based on past and current data.

The value of such models lies in forecasting, allowing us to project future trends and make informed decisions. In the internet user model, the exponential growth function has been implemented to determine future values for a specific time. This usage involves carefully determining constants based on different data points to ensure accuracy where possible.

However, it's crucial to remember that models have limitations. They can offer accurate descriptions over certain ranges but may need refinement as conditions or parameters change. For instance, as more users are connected to the internet, the model's assumptions of continuous growth may need revising when hitting natural limits such as total population. Thus, mathematical modeling remains a dynamic field, reinforcing the need for continuous learning and adaptation.