Problem 75 - A Graphical Approach to College Algebra

Question

Solve each problem. Population Growth 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 maximum population will be 51,900 in 5 years.

Step 1: Determine Initial Population

The initial population of the city five years ago was given as 49,000.

Step 2: Find Population Growth Per Year

The population increases by 580 individuals each year according to the zoning commission.

Step 3: Calculate Population Increase Over 5 Years

To find the total increase over the next 5 years, multiply the yearly increase by 5: \(580 \times 5 = 2,900\).

Step 4: Add Population Increase to Initial

Add the population increase over 5 years to the current population to find the population five years from now: \(49,000 + 2,900 = 51,900\).

Key Concepts

Algebraic Problem Solving

Algebraic problem solving is a method of finding solutions to mathematical problems using algebraic expressions and equations. This approach often involves defining variables to represent unknown quantities and then establishing relationships between these variables through equations. In the context of population growth, we're using algebraic methods to determine future population numbers.

When given a problem like the one in this exercise, you define the known and unknown information. For example, we know the population increased by 580 people each year. The total population increase over a set period forms part of the algebraic problem-solving process. You then calculate the increase using multiplication and addition to find the result. In this specific problem, using algebra helps verify that the approach to predicting future population was applied correctly.

Linear Patterns

Linear patterns are sequences of numbers that change by the same amount each time, which is known as the common difference. These patterns are called 'linear' because if graphed, they form a straight line.

In the case of the population growth problem, we'll identify the pattern by recognizing that the population increases by a fixed amount each year: 580 people. This regular increment represents a real-world linear pattern, where the population steadily climbs every year. Linear patterns like this one are critical in predicting outcomes in various fields, including demography, economics, and environmental science. Understanding linear patterns can give insights into future trends without needing to know the exact situation at every single point in time.

Arithmetic Sequence

An arithmetic sequence is an ordered list of numbers in which the difference of consecutive terms is constant. This difference is known as the common difference. Arithmetic sequences are a specific type of linear pattern, and they're particularly useful in solving problems involving regular increments like population growth.

The sequence in the population growth exercise is determined by the number of people being added each year. Given the initial population, each subsequent term is obtained by adding 580, the common difference, to the previous term. The arithmetic sequence here starts at 49,000 and increases by 580 annually. We determine the population after a number of years by finding the appropriate term in this sequence. This approach allows us to easily calculate the population 5 years from the initial point, demonstrating the power of recognizing and using arithmetic sequences in punctual problem-solving.