Problem 22 - Calculus and its applications

Question

In \(2009,\) U.S. per capita personal income \(I\) was \(\$ 48,040 .\) In \(2012,\) it was \(\$ 52,430 .\) (Source: data.worldbank.org.) Assume that the growth of U.S. per capita personal income follows an exponential model a) Letting \(t=0\) be 2009 , write the function. b) Predict what U.S. per capita income will be in 2020 . c) In what year will U.S. per capita income be double that of \(2009 ?\)

Step-by-Step Solution

Answer

a) \(I(t) = 48,040 e^{0.029t}\); b) In 2020, \(\$64,183.26\); c) Income will double in 2033.

Step 1: Define the exponential growth model

The exponential growth model is defined by the function \( I(t) = I_0 e^{kt} \), where \( I(t) \) is the income at time \( t \), \( I_0 \) is the initial income at \( t = 0 \), and \( k \) is the growth rate. Here, \( I_0 = 48,040 \).

Step 2: Determine the growth rate

Using the information for 2012 (where \( t = 3 \)), we have the equation \( 52,430 = 48,040 e^{3k} \). Solving for the rate \( k \):\[ e^{3k} = \frac{52,430}{48,040} \]Taking the natural logarithm of both sides gives:\[ 3k = \ln\left(\frac{52,430}{48,040}\right) \]Solving for \( k \), we find \( k \approx 0.029 \).

Step 3: Write the function based on the model

Inserting the value of \( k \) into the exponential model:\[ I(t) = 48,040 e^{0.029t} \]

Step 4: Predict the income in 2020

To find the income in 2020, where \( t = 11 \) (since 2020 is 11 years after 2009), substitute \( t \) into the function:\[ I(11) = 48,040 e^{0.029 \times 11} \]Calculate to obtain \( I(11) \approx \$64,183.26 \).

Step 5: Determine when the income will double

To find when income doubles, set \( I(t) = 2 \times 48,040 \) and solve for \( t \):\[ 96,080 = 48,040 e^{0.029t} \]Solving for \( t \):\[ e^{0.029t} = 2 \]\[ 0.029t = \ln(2) \]\[ t \approx \frac{0.693}{0.029} \approx 23.9 \]Hence, \( t = 24 \) (rounding 23.9 to the nearest year), which means the income will double in year 2033.

Key Concepts

Per Capita Income

In simple terms, per capita income is the average income earned per person in a particular area within a specific time. This measurement helps us understand the economic well-being of a population. To calculate per capita income, we take the total national income and divide it by the population size.
Per capita income can provide insights into the living standards and economic progress of a society, and it varies greatly across different countries and regions. When per capita income rises, it often suggests that an economy is growing and individuals might have better access to goods and services.
It's a key indicator used by economists, policymakers, and researchers to evaluate and compare the wealth of different countries or regions.

Growth Rate Calculation

Understanding how to calculate the growth rate is crucial in modeling per capita income growth. In the context of our exercise, the growth rate is a measure of how fast the per capita income increases over time. This is represented by the symbol "k" in the exponential growth model formula.
The formula for exponential growth is given by:

• \( I(t) = I_0 e^{kt} \),
• where \( I_0 \) is the initial amount (in this case, income in 2009),
• and \( k \) is the growth rate.

To find the growth rate, you can use values from different points in time. For example, between 2009 and 2012 in the exercise, by rearranging the formula based on income figures, we calculate "k" and determine it to be approximately 0.029, or 2.9%. This growth rate is vital for predicting future incomes.

Function Prediction

Predicting future values using the exponential model involves replacing the time variable t with the desired future year to estimate the outcome. This is a powerful tool, as it allows us to foresee how variables like income might evolve over time.
For instance, to predict the U.S. per capita income for the year 2020, we recognize that 2020 is 11 years after 2009, so we substitute \( t = 11 \) into our model:

• \( I(11) = 48,040 e^{0.029 imes 11} \),
• which results in an estimated income of approximately \$64,183.26.

This method not only helps in forecasting income but can also be applied to other growth patterns. Understanding function prediction is vital for making informed economic forecasts and decisions.