Problem 63 - Algebra 2

Question

Ms. Schifflet is writing a computer program to find the salaries of her employees after their annual raise. The percent of increase is represented by \(p\) . Marty's salary is \(\$ 23,450\) now. Write a polynomial to represent Marty's salary in one year and another to represent Marty's salary after three years. Assume that the rate of increase will be the same for each of the three years.

Step-by-Step Solution

Answer

The salary after one year is \( 23450(1 + \frac{p}{100}) \), and after three years is \( 23450(1 + \frac{p}{100})^3 \).

Step 1: Understand the Problem

We are tasked with creating polynomials to represent Marty's salary in one year and after three years given his salary of \( \$23,450 \) and an annual percentage increase represented by \( p \).

Step 2: Define Variables and Understand Increase Formula

Marty's current salary is \( S = 23,450 \). The annual increase is \( p\% \). So, the salary after one year is computed as: \( S' = S \times (1 + \frac{p}{100}) \). We need to simplify this formula.

Step 3: Write a Polynomial for One-Year Salary

Substitute the expression into the polynomial for one year's salary: \[ S' = 23450 \times (1 + \frac{p}{100}) \]Let \( x = \frac{p}{100} \), then after one year, salary is:\[ S' = 23450 \times (1 + x) \]

Step 4: Write a Polynomial for Three-Year Salary

Consider the polynomial for year two:\[ S'' = S' \times (1 + x) = 23450 \times (1 + x)^2 \]For three years, continue the pattern:\[ S''' = S'' \times (1 + x) = 23450 \times (1 + x)^3 \]Thus, the polynomial for Marty's salary after three years is:\[ 23450 \times (1 + \frac{p}{100})^3 \]

Step 5: Finalize the Polynomials

The polynomial for one year is:\[ 23450 \times (1 + \frac{p}{100}) \]The polynomial for three years is:\[ 23450 \times (1 + \frac{p}{100})^3 \]

Key Concepts

Annual Salary Increase

An annual salary increase is a common practice where an employee's salary is reviewed and raised once every year. This boost in pay is essential as it helps the salary keep pace with inflation and the cost of living. In the context of the exercise, Marty's salary increase is determined by a constant percentage, denoted as \(p\). Understanding this concept is crucial for calculating how much salary growth occurs over a specified period.

To determine the new salary after an annual increase, we use the formula:

• \( S' = S \times (1 + \frac{p}{100}) \)

Here, \(S\) is the base salary, \(p\) represents the percent increase, and \(S'\) gives us the salary after one year. This formula embodies how a salary increment happens annually, emphasizing the effect of a steady increase rate each year.

Percent Increase

The percent increase refers to the rate at which a salary, cost, or other numerical value grows over time. It is expressed as a percentage and is crucial in determining how much an employee’s salary grows each year. In Marty's case, the percent increase is given by \(p\), representing how much his salary should increase every year in relation to his current salary.

Calculating a percent increase involves converting the percentage to a decimal by dividing it by 100, then multiplying it with the initial amount. This process allows us to determine the exact raise amount. For example, if Marty's salary increase percent is unknown, we still represent it symbolically as \( \frac{p}{100} \), enabling us to simplify calculations effectively. This concept is key for creating accurate polynomial representations in salary projections.

Salary Growth Over Time

Salary growth over time can be modeled using polynomials, especially when dealing with consistent annual increases like those in the exercise. Here, we explored how to project Marty's salary over one and three years, assuming a stable percent increase each year.

Using a polynomial to express salary growth shortens the process for repetitive calculations. For Marty's salary over three years:

• Start with the salary for one year: \( 23450 \times (1 + \frac{p}{100}) \)
• Extend this to two years: \( 23450 \times (1 + \frac{p}{100})^2 \)
• Finally, for three years: \( 23450 \times (1 + \frac{p}{100})^3 \)

The power of the polynomial notation is its ability to streamline calculating future salaries by applying a recursive pattern, which maintains simplicity and precision over multiple periods. This formula elegantly handles repeated computations, thereby illustrating salary growth over time in a comprehensive manner.