Problem 54 - Calculus and its applications

Question

Average population. The population of the United States can be approximated by $$ P(t)=282.3 e^{0.01 t} $$ where \(P(t)\) is in millions and \(t\) is the number of years since 2000. (Source: Population Division, U.S. Census Bureau.) Find the average value of the population from 2009 to 2013.

Step-by-Step Solution

Answer

The average population from 2009 to 2013 is approximately 315.14 million.

Step 1: Understand the function

The given function for the population, \( P(t) = 282.3 e^{0.01t} \), models how the population in millions changes over time, where \( t \) is the number of years since 2000.

Step 2: Set the integration limits

We need to find the average population from 2009 to 2013. Thus, the limits for \( t \) are from \( t=9 \) (for 2009) to \( t=13 \) (for 2013).

Step 3: Formula for average value of a function

The average value of a continuous function \( f(t) \) from \( a \) to \( b \) is given by \( \frac{1}{b-a} \int_a^b f(t)\, dt \). Here, \( f(t) = 282.3 e^{0.01t} \), \( a = 9 \), and \( b = 13 \).

Step 4: Integrate the function

Compute \( \int_9^{13} 282.3 e^{0.01t} \, dt \). Integrating, we have:\[ \int 282.3 e^{0.01t} \, dt = \frac{282.3}{0.01} e^{0.01t} = 28230 e^{0.01t} \]Therefore, \( \int_9^{13} 282.3 e^{0.01t} \, dt = 28230 [e^{0.13} - e^{0.09}] \).

Step 5: Calculate the definite integral

Calculate \( 28230 [e^{0.13} - e^{0.09}] \). Using the approximate values of \( e^{0.13} \approx 1.13883 \) and \( e^{0.09} \approx 1.09417 \), we have:\[ 28230 (1.13883 - 1.09417) = 28230 \times 0.04466 \approx 1260.56 \]

Step 6: Compute the average value

The average population value from 2009 to 2013 is given by the formula: \( \frac{1}{13-9} \int_9^{13} 282.3 e^{0.01t} \, dt \).Thus,\[ \frac{1260.56}{4} = 315.14 \] million.

Key Concepts

Exponential Growth Model

An exponential growth model is widely used to describe scenarios where quantities grow at rates proportional to their current value. In such models, the rate of change increases over time, leading to exponential increases.
The function given in the exercise is an example of an exponential growth model:
\[ P(t) = 282.3 e^{0.01t} \]
Here, \( P(t) \) represents the population in millions, and \( e^{0.01t} \) signifies the exponential growth over time.

• **Base Value:** The initial population value in 2000 is 282.3 million.
• **Growth Rate:** The rate, 0.01, indicates a 1% increase per year.
• **Time Variable (t):** Defines years since 2000, making calculations or predictions for any specific year straightforward.

Exponential models serve as an essential tool in a variety of fields, allowing predictions of population changes, economic growth, and even cell proliferation in biological studies.

Definite Integral Calculation

Calculating a definite integral provides the accumulated value of a function over a specific interval. It is key to finding areas under curves, which is essential in determining the total change over time.
Here, the definite integral helps in finding the total increase in population from 2009 to 2013. The integral of the given function over the interval from \(t = 9\) (2009) to \(t = 13\) (2013) is:
\[ \int_9^{13} 282.3 e^{0.01t} \, dt \]
The process involves integrating the function, resulting in:
\[ \frac{282.3}{0.01} e^{0.01t} = 28230 e^{0.01t} \]
After evaluating from \( t = 9 \) to \( t = 13 \), we substitute the values into the antiderivative to find:

• Substitute and solve: \( 28230 [e^{0.13} - e^{0.09}] \)
• Approximate values: \( e^{0.13} \approx 1.13883 \) and \( e^{0.09} \approx 1.09417 \)
• Calculate: \( 28230 \times 0.04466 \approx 1260.56 \)

This integral gives the accumulated population growth in millions over the observed period.

Population Modeling

Population modeling is a method of creating mathematical representations to predict how a population will grow over time. These models help governments, researchers, and organizations make informed decisions by forecasting future trends.
In the exercise, the model \( P(t) = 282.3 e^{0.01t} \) describes the expected population growth in the U.S.
Population models, like the one used, capture several critical aspects:

• **Initial Conditions:** The starting population size, which anchors future predictions.
• **Growth Dynamics:** The constants and functions reflecting how and at what rate the population is expected to grow.
• **Predictive Power:** Ability to project future population sizes and necessary resources more accurately.

This model helps estimate the average population from 2009 to 2013 by integrating over the interval \([9, 13]\), then dividing by the interval length, providing valuable information for planning in fields like urban development, environmental management, and policy formulation.