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Chapter 4: Problem 14

Mexico Population (Historic) The population of Mexico between 1921 and 2010 can be modeled as $$ p(t)=8.028 e^{0.025 t} \text { million people } $$ where \(t\) is the number of years since 1900 . (Source: Based on data from Statistical Abstract, 2009 and www.inegi.gob.mx) a. What was the population of Mexico and how rapidly was it growing in \(2010 ?\) b. Write a linearization of \(p\) in 2010 . c. Use the linearization to estimate the population of Mexico in 2011 .

Short Answer

Expert verified
In 2010, the population was 125.512 million, growing at 3.14 million/year. Estimated 2011 population is 128.652 million using linearization.

Step by step solution

01

Calculate Population for 2010

To determine the population of Mexico in 2010, we need to plug in the value of \( t = 110 \) (because 2010 is 110 years after 1900) into the function \( p(t) = 8.028 e^{0.025 t} \). Calculate:\[p(110) = 8.028 e^{0.025 \times 110} \approx 8.028 e^{2.75} \approx 8.028 \times 15.6426 \approx 125.512 \text{ million people}\]Thus, the population in 2010 was approximately 125.512 million people.
02

Calculate Growth Rate for 2010

The growth rate is found by differentiating the population function \( p(t) = 8.028 e^{0.025t} \) concerning \( t \). The derivative is:\[p'(t) = 8.028 \cdot 0.025 \cdot e^{0.025t} = 0.2007 e^{0.025t}\]Evaluate this derivative at \( t = 110 \):\[p'(110) = 0.2007 e^{0.025 \times 110} = 0.2007 \times 15.6426 \approx 3.14 \text{ million people per year}\]Thus, in 2010, the population was growing at approximately 3.14 million people per year.
03

Linearize the Population Function at 2010

Linearization uses the tangent line approximation at a point. For the function \( p(t) \), the linearization at \( t = 110 \) is:\[L(t) = p(110) + p'(110)(t - 110)\]Using values from previous calculations:\[L(t) = 125.512 + 3.14(t - 110)\]
04

Use Linearization to Estimate 2011 Population

Use the linearization formula to estimate the population in 2011, where \( t = 111 \):\[L(111) = 125.512 + 3.14 \times (111 - 110) = 125.512 + 3.14 = 128.652 \text{ million people}\]Therefore, using linearization, the estimated population of Mexico in 2011 was approximately 128.652 million people.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Modeling
Population modeling is the process of creating mathematical formulas to represent how a population changes over time. These models help predict future population sizes and analyze trends.

In the provided example, the population of Mexico between 1921 and 2010 is modeled using an exponential growth formula. The formula is given as \( p(t) = 8.028 \times e^{0.025t} \), where \( t \) is the number of years since 1900. This exponential function reflects continuous growth, meaning the population increases at a rate proportional to its current size.

Exponential population models are particularly useful for understanding how quickly populations grow and making projections for the future. However, they assume that factors like birth rate and migration remain constant, which might not always reflect real-world conditions. Nevertheless, such models provide an excellent foundation for analyzing population dynamics over time.
Linearization
Linearization is a mathematical technique used to approximate a non-linear function by a linear one, usually around a specific point. This method simplifies complex calculations and allows for easier analysis of how small changes affect the function.

Linearization is particularly valuable when dealing with functions that are difficult to work with in their original form. For the population model of Mexico, the function \( p(t) = 8.028 e^{0.025t} \) is non-linear. By linearizing it at \( t = 110 \) (the year 2010), we can find an approximation of the function that is easier to work with, especially for close values of \( t \).

The linearization involves calculating the tangent line at the point of interest. Mathematically, it's expressed as \( L(t) = p(110) + p'(110)(t - 110) \), where \( p(110) \) and \( p'(110) \) are the function and its derivative evaluated at \( t = 110 \), respectively. This linear function closely approximates the population for values of \( t \) near 110.
Derivative
The derivative is a central concept in calculus, reflecting how a function changes at any given point. In simpler terms, a derivative gives the rate of change or "slope" of the function. It can be applied to various contexts, such as finding how quickly populations grow, as in this exercise.

For the population model \( p(t) = 8.028 e^{0.025t} \), the derivative \( p'(t) = 0.2007 e^{0.025t} \) measures how fast the population is growing at any time \( t \). Calculating \( p'(110) \) provided the growth rate in 2010, which was approximately 3.14 million people per year.

This growth rate is crucial for making short-term predictions and understanding current trends. Through derivatives, we can derive insights such as whether the population is growing faster or slower than in previous years, and how long it might take to reach certain population milestones. Such analyses are vital for planning resources, infrastructure, and policies.

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