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Chapter 3: Problem 4

Consider the following job offers. At Acme Corporation, you are offered a starting salary of \(\$ 20,000\) per year, with raises of \(\$ 2500\) annually. At Boca Corporation, you are offered \(\$ 25,000\) to start and raises of \(\$ 2000\) annually. a. Find an equation to represent your salary, \(S_{A}(n),\) after \(n\) years of employment with Acme. b. Find an equation to represent your salary, \(S_{B}(n),\) after \(n\) years of employment with Boca. c. Create a table of values showing your salary at each of these corporations for integer values of \(n\) up to 12 years. d. In what year of employment would the two corporations Day you the same salary?

Short Answer

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Step by step solution

01

Calculate the salary equation for Acme Corporation

Acme Corporation offers a starting salary of \(\$ 20,000\), with raises of \(\$ 2,500\) every year. To write the equation, note that the starting salary plus annual raises can be modeled as a linear function. The general form of the salary equation for Acme Corporation after \(n\) years is given by: \(S_{A}(n) = 20000 + 2500n\)
02

Calculate the salary equation for Boca Corporation

Boca Corporation offers a starting salary of \(\$ 25,000\), with raises of \(\$ 2,000\) every year. Using a similar linear function format, the salary equation for Boca Corporation after \(n\) years is: \(S_{B}(n) = 25000 + 2000n\)
03

Create a table of values

Using the equations obtained in Steps 1 and 2, create a table with values of \(n\) from 0 to 12 to compare the salary at both corporations. | Years (n) | Acme Salary (\

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Functions
Linear functions play a key role in salary comparison exercises. A linear function is a type of mathematical function that graphs as a straight line. The general form of a linear equation is represented as \( y = mx + b \), where \( y \) is the dependent variable, \( m \) is the slope (rate of change), \( x \) is the independent variable, and \( b \) is the y-intercept (the starting value).
In the context of our salary comparison, the salary equations are linear functions: \( S_{A}(n) = 20000 + 2500n \) for Acme Corporation and \( S_{B}(n) = 25000 + 2000n \) for Boca Corporation.
Here:
  • The starting salaries (\( 20000 \) for Acme and \( 25000 \) for Boca) act as the y-intercept \( b \).
  • The annual raises (\( 2500 \) for Acme and \( 2000 \) for Boca) represent the slope \( m \).
  • The number of years \( n \) is the independent variable \( x \).
Annual Raises
Annual raises are increments in salary that employees receive at regular intervals, typically every year. In our exercise, Acme Corporation offers an annual raise of \( 2500 \) dollars, whereas Boca Corporation offers \( 2000 \) dollars.
Here's how these raises influence the salary over time:
  • The annual raise for Acme means that for every additional year, the salary increases by \( 2500 \) dollars.
  • Likewise, the annual raise for Boca means an increase of \( 2000 \) dollars each year.
These raises transform the starting salary into a linear function, demonstrating a steady, predictable growth over time depending on the number of years served. This steady increase can be modeled mathematically and visualized graphically as a straight line.
Table of Values
Creating a table of values involves calculating the salary at each year for both corporations. This helps visualize the growth in salary over time and facilitates a comparison between the two offers.
Here鈥檚 how you create a table of values for up to 12 years:
  • Determine the starting salary for each corporation (\( 20000 \) for Acme and \( 25000 \) for Boca).
  • Add the annual raise for each subsequent year (\( 2500 \) for Acme and \( 2000 \) for Boca).
  • For example, after 5 years, the salary at Acme would be \( 20000 + 2500 \times 5 = 32500 \), and at Boca, it would be \( 25000 + 2000 \times 5 = 35000 \).
By calculating these values year-by-year, you can construct a comparison table and clearly see how the salaries evolve over time.
Equality of Salaries
To determine the year when both corporations pay the same salary, we need to find when \( S_{A}(n) = S_{B}(n) \). Essentially, we set the two salary equations equal to one another and solve for \( n \): \[ 20000 + 2500n = 25000 + 2000n \].
Manipulating the equation, we consolidate terms on one side: \[ 20000 + 2500n - 2000n = 25000 \], which simplifies to \[ 500n = 5000 \], and thus \[ n = 10 \].
This result tells us that after 10 years of employment, both corporations would offer the same salary. Understanding the equality of salaries through solving equations is crucial for making informed decisions regarding job offers based on long-term financial outcomes.

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