Question

Use either the formulas or the universal growth and decay curves, as directed by your instructor.If the population of a country was 11.4 million in 2010 and grows at an annual rate of \(1.63 \%,\) find the population by the year \(2015 .\)

Short answer

The population by the year 2015 is approximately 12.3594 million.

Step-by-step solution

Understand the Problem

We need to calculate the population of a country in the year 2015, given that it was 11.4 million in 2010 and that it grows at an annual rate of 1.63%.

Use the formula for exponential growth

The formula for exponential growth is given by \(P(t) = P_0 \times (1 + r)^t\) where \(P(t)\) is the population after time \(t\), \(P_0\) is the initial population, \(r\) is the growth rate, and \(t\) is the time in years.

Plug in the given values

Here, \(P_0 = 11.4\) million, \(r = 0.0163\), and \(t = 2015 - 2010 = 5\) years. So, plug these values into the formula: \(P(5) = 11.4 \times (1 + 0.0163)^5\).

Calculate the population in 2015

Perform the calculation: \(P(5) = 11.4 \times (1 + 0.0163)^5 = 11.4 \times 1.0841 = 12.3594\) million. Therefore, the population in 2015 is approximately 12.3594 million.

Key concepts

Population Growth

Understanding population growth is key to predicting changes in the numbers of inhabitants in a particular area over time. The term refers to the increase in the number of individuals in a population. Common factors influencing population growth include birth rates, death rates, and migration. In our daily lives, we observe population growth when cities become more crowded or when a species' number increases in an ecosystem.

Population growth can be tracked using data from past years and can be modeled mathematically to make predictions for future years. In the given exercise, we have the task of calculating the expected population of a country from an initial value and a specific growth rate. This real-life example highlights the importance of understanding mathematical models for decision-making in urban planning, environmental assessment, and resource management.

Exponential Growth Formula

Exponential growth occurs when the increase in a population's size is proportional to the current size, resulting in the population growing at an increasing rate over time. It's important to understand that exponential growth cannot continue indefinitely due to real-world constraints, such as limited resources.

The formula for exponential growth, as used in the step-by-step solution, is a mathematical representation of how populations can grow under ideal, unlimited conditions. It is expressed as:
\[ P(t) = P_0 \times (1 + r)^t \]
where:\[ P(t) \] is the population at time \( t \), \[ P_0 \] is the initial population, \( r \) is the growth rate (expressed as a decimal), and \( t \) is the time in years. By using this formula, we can calculate the expected population size after a certain period, assuming a consistent growth rate. This aids in various planning and policy-making processes where population forecasts are necessary.

Mathematical Modeling

Mathematical modeling is the process of using mathematical expressions and equations to represent real-world situations. In the context of population growth, mathematical models allow us to predict future changes in population sizes with reasonable accuracy.

Models like the exponential growth formula can be simple yet powerful tools that policymakers, scientists, and businesses use to make informed decisions. However, it's important to note that models are approximations and their predictions are conditional on the assumption that current trends continue. For instance, a sudden change in the birth rate, death rate, or migration patterns could alter the projected population size. Therefore, while mathematical models are essential, they should be used in conjunction with real-world observations and continuous data updating.

Exponential Functions

Exponential functions are mathematical functions that describe situations where a quantity grows or decays at a rate proportional to its current value. These functions are of the form \( f(x) = a \times b^x \), where \( a \) represents the initial amount, \( b \) is the base representing the growth (if greater than 1) or decay (if between 0 and 1) rate, and \( x \) is the exponent, often representing time.

In the exercise provided, the exponential function helps us understand how populations change over time. Exponential functions are used in a wide range of applications, from compound interest calculations in finance to radioactive decay in physics, due to their unique properties. Notably, these functions increase rapidly, which is why they are often associated with scenarios like population booms and the spread of diseases in epidemiology.