Problem 57 - Algebra and Trigonometry

Question

Explaining the Concepts Nigeria has a growth rate of 0.025 or \(2.5 \% .\) Describe what this means.

Step-by-Step Solution

Answer

Nigeria's growth rate of 0.025 or 2.5% means that every year, the population of Nigeria increases by 2.5% of the population at the start of that year.

Step 1: Understand the Concept of Growth Rate

A growth rate, in the simplest terms, is a measure of how something changes over time as a percentage of the original. In this case, it applies to the population of Nigeria.

Step 2: Express Growth Rate in Real Terms

When we say Nigeria has a growth rate of 0.025 (or 2.5%), it means that every year the population of Nigeria increases by 2.5% of the population at the start of that year.

Step 3: Apply the Concept

So if, for example, Nigeria had a population of 100,000 at the start of a year, by the end of that year the population would have increased by 2,500 (which is 2.5% of 100,000).

Key Concepts

Mathematical Representation of Growth

When discussing the concept of growth rate, particularly in relation to population, a precise mathematical representation is crucial for both understanding and calculation purposes. Imagine a petri dish with a few bacteria that double every few hours. Similarly, human populations grow over time, though at a much slower rate. The growth rate is expressed as a percentage which represents the rate of change over a time period—usually one year for populations.

In mathematical terms, the population growth rate (PGR) can be represented using the formula:\[\begin{equation}\textrm{PGR} = \frac{\textrm{Population at end of period} - \textrm{Population at start of period}}{\textrm{Population at start of period}} \times 100\%\end{equation}\]
For instance, in our Nigeria example, the 2.5% growth rate can be interpreted using this formula by understanding that the 'Population at end of period' is increased by 2.5% from the 'Population at start of period' due to the growth. To simplify, if Nigeria's population at the start of the year is represented as P, by the end of the year it will be P + (0.025 * P).

Percentage Growth Calculation

Percentage growth calculation takes the abstract concept of growth and applies it to real numbers. By relating the growth rate to the actual starting figure, we get a way to concretely understand the real-world change.

To calculate the percentage growth, you multiply the starting value by the growth rate. For example, consider a university with 10,000 students that increases its student body by 5% annually. To calculate the number of new students, we take 10,000 and multiply it by 0.05 (5% in decimal form), giving us 500 new students for that year.

Scenario Application

When applying this to the Nigeria population example, if the starting population is 100,000, the population growth is calculated as follows:\[\begin{equation}100,000 \times 0.025 = 2,500\end{equation}\]So, Nigeria's population would grow by 2,500 people over that year. This method is vital in numerous fields, such as demography, economics, and biology, offering a clear insight into growth trends over time.

Applying Growth Rates in Real-World Scenarios

Applying growth rates in real-world scenarios allows governments, institutions, and individuals to plan for the future. For example, city planners use population growth rates to forecast the need for housing, infrastructure, and services. Economists use growth rates to predict economic expansion or contraction, while businesses use them to forecast sales and scale operations.

In our Nigeria example, a growth rate of 2.5% can lead to various real-world considerations. If the government knows the growth rate, it can plan for infrastructure such as schools and hospitals, ensure sufficient food supply, and manage natural resources responsibly. It's also important for setting policies on immigration, health, and social security.

Long-term Implications

The long-term implications of sustained growth rates can be profound. In a society where the population grows exponentially, resources may become scarce, and the environment may be stressed. Conversely, a declining growth rate might indicate aging populations and lead to different societal challenges. Understanding the mathematical and practical implications of growth rates paves the way for sustainable development strategies.