Problem 83 - A Graphical Approach to Precalculus with Limits

Question

Five years ago, the population of a city was \(49,000 .\) Each year, the zoning commission permits an increase of 580 in the population. What will the maximum population be 5 years from now?

Step-by-Step Solution

Answer

The population will be 54,800 in 5 years.

Step 1: Determine Initial Population

Five years ago, the population was 49,000. To find the current population, we need to account for the yearly increase over the past 5 years.

Step 2: Calculate Current Population

Since the population increased by 580 each year for the past 5 years, we calculate the current population by adding \(580 \times 5\) to the population from five years ago. \[ 49,000 + (580 \times 5) = 49,000 + 2,900 = 51,900 \] So, the current population is 51,900.

Step 3: Calculate Population in Next 5 Years

Similarly, the population will continue to increase by 580 each year for the next 5 years. Add \(580 \times 5\) to the current population.\[ 51,900 + (580 \times 5) = 51,900 + 2,900 = 54,800 \]Thus, the population 5 years from now will be 54,800.

Key Concepts

Population Growth

Population growth is a key concept in understanding how populations increase over time, especially in a controlled environment like a city. In this context, population growth refers to the systematic increase in the number of residents in a city, often influenced by factors such as birth rates, zoning permissions, and migrations. Here, the exercise focuses on a controlled population increase determined by zoning regulations.

When calculating population growth, it's essential to start with the initial population size and then apply the annual increase rate over the specified period. This exercise exemplified this by starting from a population of 49,000 people and adding a consistent 580 people per year.

This form of arithmetic growth is linear because each year adds the same number of people to the population, which contrasts with exponential growth where the increase becomes faster over time.

Linear Functions

Linear functions are mathematical equations that create straight lines when graphed on a coordinate plane. The formula for a linear function is typically written as \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) is the y-intercept.

In this exercise about population growth, you essentially have a linear function without a direct graph representation. The population increase can be described by the linear function \(P(x) = 580x + 49,000\), where \(x\) represents the number of years since the initial population measurement.

Here, the slope (580) indicates the yearly increase in population, and the y-intercept (49,000) is the original population size. Linear functions are quite common and useful in real-world applications, such as predicting growth trends in urban planning.

Word Problems

Word problems can sometimes be challenging because they require translating a real-world scenario into mathematical expressions. This involves understanding the problem's context, identifying the relevant numerical information, and determining the mathematical operations needed to solve the problem.

In the given population growth problem, we are asked to determine the population in future years based on a set annual increase. To tackle a word problem like this one, it's important to:

• Identify the known data (e.g., initial population, annual increase)
• Define what is being solved (e.g., future population after a certain number of years)
• Apply the correct mathematical operations to find the solution

This structured approach to solving word problems ensures that you systematically work through all aspects of the problem and arrive at a logical solution.