Problem 1 - College Algebra

Question

The exponential models describe the population of the indicated country, \(A,\) in millions, \(t\) years after \(2010 .\) Use these models to solve Exercises \(1-6\) Indis \(\quad A=1173.1 e^{0.005 t}\) Ing \(\quad A=31.5 e^{0.018}\) Japer \(\quad A=127.3 e^{-0.006 t}\) Restis \(\quad A=141.9 e^{-0.005 t}\) What was the population of Japan in \(2010 ?\)

Step-by-Step Solution

Answer

The population of Japer (Japan) in 2010 was 127.3 million.

Step 1: Identify Relevant Exponential Model

Among the given country models, identify the one used for Japer, which is \(A = 127.3 \cdot e^{-0.006 t}\).

Step 2: Apply The Correct Timeframe

From the problem statement, the year for which we are interested in the population is 2010. Since \(t\) represents the number of years after 2010, that makes \(t = 0\).

Step 3: Solve Using The Population Model

Plug in \(t = 0\) into the model to find the population for the year 2010. So it will be \(A = 127.3 \cdot e^{-0.006 \cdot 0} = 127.3 \cdot e^0\). Since any number to the power of 0 is 1, that gives \(A = 127.3 \cdot 1\).

Step 4: Evaluate The Result

The resulting population is \(A = 127.3\) million. This implies that Japer's population was 127.3 million in 2010. Remember, the final answer should be in million, as the model describes the population in millions.

Key Concepts

Population Growth

Population growth refers to the increase in the number of individuals in a population. It is a fundamental concept in ecology, economics, and planning. Understanding how populations grow is crucial for various applications, including resource management, urban development, and environmental conservation.

There are different models to describe population growth, with the simplest being the linear model. However, many biological populations tend to grow exponentially, especially when resources are abundant and the population is far from its carrying capacity. An exponential growth pattern means the population size increases at a rate proportional to its current size, leading to faster and faster growth as the population gets larger.

Implementing population growth models to real-world scenarios, such as forecasting the future size of a city or anticipating the spread of a species, is an invaluable tool, allowing policy-makers, scientists, and stakeholders to make informed decisions.

Exponential Growth Equation

The exponential growth equation is a mathematical representation of how a quantity grows over time at a rate that is proportional to its current value. In the context of populations, it is typically written as: \[ P(t) = P_0 \times e^{rt} \],where \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population size, \( e \) is the base of the natural logarithm (approximately equal to 2.71828), and \( r \) is the growth rate.

This equation is used to calculate the size of a population at any point in the future, given its current size and growth rate. The exponential nature of the equation means that small changes in the growth rate can lead to large differences in population size over time. Understanding this equation allows students to solve problems involving population growth, as in the given exercise, by adjusting the variables as needed.

Population Model Application

Applying population models, like the exponential growth equation, to real-world situations allows us to predict and analyze trends in population dynamics. For instance, in the textbook exercise provided, the models given help in understanding the growth or decline of countries' populations over time.

To apply these models effectively, it is essential to accurately insert values for parameters, such as the initial population size and growth rate, which can be derived from empirical data or estimates. In the Japer case, the initial population (\( P_0 \)) is given as 127.3 million with a decline rate of 0.006 per year (since the rate is negative). By substituting these values into the exponential growth equation with time set to zero, the current population is easily found.

Real-life applications of population models extend far beyond academic exercises. They are used in public health for predicting the spread of diseases, in environmental science for species conservation, and in urban planning for infrastructure development. The ability to apply these models helps to create strategies for sustainable management and growth, which is critical in our ever-changing world.