Problem 36 - College Algebra Essentials

Question

Use the formula \(t=\frac{\ln 2}{k}\) that gives the time for a population with a growth rate \(k\) to double to solve Exercises \(35-36 .\) Express each answer to the nearest whole year. The growth model \(A=112.5 e^{0.012 y}\) describes Mexico's population, \(A,\) in millions, \(t\) years after 2010 . a. What is Mexico's growth rate? b. How long will it take Mexico to double its population?

Step-by-Step Solution

Answer

a. The growth rate of Mexico is \(0.012\). b. It will take approximately 58 years for Mexico to double its population.

Step 1: Calculate the growth rate

In the provided population growth model \(A=112.5 e^{0.012 y}\), the growth rate \(k\) is the coefficient of \(y\) in the exponent of \(e\), which is \(0.012\). So Mexico's growth rate is \(0.012\).

Step 2: Apply the doubling time formula

The formula for the doubling time of a population is \(t=\frac{\ln 2}{k}\). By substituting the growth rate \(k=0.012\) into the formula, we get \(t = \frac{\ln 2}{0.012}\).

Step 3: Calculate the doubling time

Finally, calculate the doubling time using a calculator in order to get a numerical answer that should be rounded off to the nearest whole year. It should look something like \(t = \frac{0.693}{0.012}\).

Step 4: Final Answer

With the given values, \(t\) should yield a result of \(t \approx 58\) years.

Key Concepts

Doubling Time Formula

The doubling time formula is a handy tool for predicting how long it takes for a population to double in size, given a constant rate of growth. It's an important concept in both mathematics and real-world applications, as it helps us understand growth dynamics.
The formula is: \[ t = \frac{\ln 2}{k} \] where:

• \(t\) is the doubling time.
• \(\ln 2\) is the natural logarithm of 2, approximately 0.693.
• \(k\) is the growth rate (expressed as a decimal).

To use it, simply insert the growth rate into the formula as \(k\). Solving for \(t\), you'll find how long it takes for the initial amount to multiply by two. This formula is straightforward, making it extremely valuable for quick calculations.

Population Growth Rate

The population growth rate is a metric that indicates how quickly a population is increasing or decreasing over time. It is usually expressed as a decimal or percentage, showing the change relative to the number of individuals.
For instance, in the growth model \(A = 112.5e^{0.012y}\), the number 0.012 represents the population growth rate. This tells us that each year, the population increases by about 1.2%.
Understanding the growth rate is essential as it forms the basis for other calculations, such as those involving exponential growth and the doubling time formula. A stable, positive rate indicates that the population is growing, whereas a negative rate would suggest a decline.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They model situations where a quantity grows or declines at a rate proportional to its current value. These functions are essential in describing phenomena such as population growth.
In the context of population growth, the function \(A = 112.5e^{0.012y}\) represents the exponential model. Here, \(e\) is the base, an irrational number approximately equal to 2.71828. The term \(e^{0.012y}\) shows exponential growth over time, where \(y\) is the time in years.
Exponential functions are characterized by their rapid increase or decrease. This makes them highly useful in modeling real-world situations where such dynamics are observed, providing a foundation for understanding and predicting population changes.

Natural Logarithm

The natural logarithm is a logarithm to the base \(e\), where \(e\) is an irrational constant roughly equal to 2.71828. In expressions, it's often written as \(\ln x\). The natural logarithm is widely used in mathematics, particularly in solving exponential growth or decay problems.
The natural logarithm serves a critical role in the doubling time formula \(t = \frac{\ln 2}{k}\). Here, \(\ln 2\) represents the time factor required for doubling, highlighted by the fact that it's always the same, regardless of the specific situation.
Understanding the natural logarithm's properties is essential for simplifying and solving complex equations involving exponential functions. It helps decode how quantities evolve over time, making it a valuable concept in both mathematical theory and practical computations.