Problem 17 - Calculus Concepts

Question

Mexico Internet Users The number of Internet users in Mexico between 2004 and 2008 can be modeled as \(u(t)=8.02\left(1.17^{t}\right)\) million users where \(t\) is the number of years since 2004 (Source: Based on data at www.internetworldstats.com/am/mx.htm) a. On average, what was the rate of change in the number of Internet users in Mexico between 2004 and \(2008 ?\) b. What was the percentage change in the number of Internet users in Mexico between 2004 and \(2008 ?\) c. The population of Mexico in 2008 was 109,955,400 . What percentage of the Mexican population used the Internet in \(2008 ?\)

Step-by-Step Solution

Answer

a) 1.78 million users/year; b) 88.65%; c) 13.76%

Step 1: Identify the function and parameters

The function given is \( u(t) = 8.02 \times 1.17^t \), which describes the number of Internet users in millions where \( t \) is the number of years since 2004. We will use this function to find required quantities for parts a, b, and c.

Step 2: Calculate the number of Internet users in 2004

Substitute \( t = 0 \) into the function since 2004 corresponds to \( t = 0 \): \[ u(0) = 8.02 \times 1.17^0 = 8.02 \text{ million users} \]

Step 3: Calculate the number of Internet users in 2008

Substitute \( t = 4 \) into the function since 2008 corresponds to \( t = 4 \): \[ u(4) = 8.02 \times 1.17^4 = 8.02 \times 1.8867 = 15.13 \text{ million users} \]

Step 4: Calculate the average rate of change (part a)

The average rate of change between 2004 and 2008 is the difference in users over the time period. So: \[ \text{Average rate of change} = \frac{u(4) - u(0)}{4 - 0} = \frac{15.13 - 8.02}{4} = \frac{7.11}{4} = 1.78 \text{ million users per year} \]

Step 5: Calculate the percentage change (part b)

The percentage change is calculated by the formula: \[ \text{Percentage change} = \left( \frac{u(4) - u(0)}{u(0)} \right) \times 100\%\]Substitute the values into the formula:\[ \text{Percentage change} = \left( \frac{15.13 - 8.02}{8.02} \right) \times 100\% = \left( \frac{7.11}{8.02} \right) \times 100\% \approx 88.65\% \]

Step 6: Calculate the internet usage percentage in 2008 (part c)

Find the percentage of the total population using the internet by dividing the number of users by the population, and then multiplying by 100: \[ \text{Percentage of population} = \left( \frac{15.13 \text{ million}}{109.9554 \text{ million}} \right) \times 100\% \approx 13.76\% \]

Key Concepts

Calculus Rate of Change

Understanding the rate of change is essential in calculus as it describes how a quantity varies over time. In our exercise, we analyzed the change in the number of Internet users in Mexico from 2004 to 2008. The function \( u(t) = 8.02 \times 1.17^t \) helps us model this scenario.

To find the "average rate of change," we look at how much the quantity changes over a specific period. By examining the number of internet users in 2004 (\( u(0) = 8.02 \) million) and 2008 (\( u(4) = 15.13 \) million), we determined the overall increase in users. The formula for the average rate of change is calculated as \( \frac{u(4) - u(0)}{4} = 1.78 \) million users per year.

This means, on average, the number of users increased by 1.78 million each year during the given period. This calculation helps businesses, governments, and researchers understand how quickly a market or phenomenon grows over time.

Percentage Change

The concept of percentage change is a useful tool to assess the relative growth or reduction in a quantity over time. It's expressed as a percentage to provide a clearer sense of magnitude.

In this exercise, percentage change is calculated to understand how the number of Internet users in Mexico grew between 2004 and 2008. We use the formula:
\[ \text{Percentage change} = \left( \frac{u(4) - u(0)}{u(0)} \right) \times 100\% \]
Inserting the values, we have:
\( \frac{15.13 - 8.02}{8.02} \times 100\% = 88.65\% \).

This result indicates that there was an 88.65% increase in the number of users, showing a significant expansion in internet usage during the analyzed period. This type of information is crucial for understanding market trends and potential.

Mathematical Modeling

Mathematical modeling plays a key role in predicting and analyzing real-world phenomena through abstract concepts. In our example, the function \( u(t) = 8.02 \times 1.17^t \) serves as a model to describe the exponential growth of Internet users in Mexico over several years.

This model uses parameters to represent initial conditions and growth factors effectively. Here, \( 8.02 \) is the initial number of users, whereas \( 1.17 \) indicates the growth rate per year. Such models allow for predictions of behavior based on existing data, demonstrating how two variables (time and number of users) are related.

By applying mathematical modeling, we gather insights not only into growth patterns but also potential future trends. Businesses and policymakers can use these insights to make informed decisions.