Problem 22 - College Algebra with Modeling and Visualization

Question

Job Offer A new employee is offered \(\$ 35,000\) per year with a guaranteed \(\$ 5000\) raise each year. Is this an example of linear or exponential growth? Find a function \(f\) that computes the salary during the \(n\) th year.

Step-by-Step Solution

Answer

Linear growth; function is \(f(n) = 35000 + 5000(n-1)\).

Step 1: Identify the Pattern

The problem states that an employee is offered \(\\( 35,000\) for the first year and receives a \(\\)5000\) raise every subsequent year. This indicates the salary increases by a constant amount each year.

Step 2: Determine the Type of Growth

Since the salary increases by a constant amount each year (\(\$5000\)), this is linear growth. Linear growth has a fixed addition, unlike exponential growth which would involve multiplication.

Step 3: Define the Function

To express the salary in mathematical terms, note that for year 1 \((n=1)\), the salary is \(\\( 35,000\). For the subsequent years, each salary can be found by adding \(\\)5000\) to the previous year's salary. The function is:\[f(n) = 35000 + 5000(n-1)\]where \(n\) is the year number.

Key Concepts

Arithmetic Sequence

Understanding arithmetic sequences can be incredibly helpful in tackling problems involving linear growth. An arithmetic sequence is a series of numbers in which each term increases by a fixed, constant amount.
In our case, the employee's salary increases by a consistent \(\\(5000\) every year. This constant increment aligns with the definition of an arithmetic sequence.
Some important features of an arithmetic sequence include:

• The first term, which is the starting point of the sequence. Here, it is \(\\)35,000\)
• The common difference, which is the constant amount added to each subsequent term. It is \(\$5000\) in our scenario.

To express an arithmetic sequence mathematically, you can use the formula:\[a_n = a_1 + (n-1)d\]Where:

• \(a_n\) is the n-th term or salary in this context.
• \(a_1\) is the first term or starting salary.
• \(n\) is the position in the sequence or the year number.
• \(d\) is the common difference or the raise amount.

Salary Function

Understanding the concept of a salary function helps map out how the compensation of an employee evolves over time. The salary function uses mathematical expressions to calculate the salary for a given year, based on identified patterns or rules.
In the exercise, we derive the salary function by observing the yearly increment pattern. The salary function captures this increase in a clear formula, as seen:\[f(n) = 35000 + 5000(n-1)\]Here \(f(n)\) represents the salary during the \(n\)th year.
Some key ideas involved in defining a salary function:

• The base salary (\(\\(35,000\)) is the starting point of our function.
• The linear growth through addition of \(\\)5000\) for each subsequent year is reflected in the expression \(5000(n-1)\).
• The variable \(n\) represents the current year being considered.

By using this function, computing the salary for any given year becomes straightforward, ensuring both employees and employers know what to expect in a clear and efficient way.

Mathematical Modeling

Mathematical modeling is a powerful tool used to represent real-world scenarios with mathematical expressions, providing a simplified framework to understand complex situations.
In this case, the employment salary dynamics are modeled with a mathematical function.
By transforming the salary increase pattern into a function, one can explore scenarios, estimate future salaries, and make informed decisions. Here, our modeling approach involves:

• Identifying the initial conditions and constants such as the first-year salary (\(\\(35,000\)) and the annual raise (\(\\)5000\)).
• Choosing a mathematical structure, recognizing that a constant annual increment suggests a linear model.
• Formulating a function that accurately represents the situation, handling infinite scenarios simply by changing the input year number \(n\).

This process greatly assists in planning and forecasting financial expectations and helps make logical predictions based on past and projected trends. Mathematical modeling thus acts as a bridge between theoretical mathematics and practical applications.