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Chapter 6: Problem 3

When Lisa started at her current job, her employer gave her two days of paid vacation time with a promise of three additional paid vacation days for each year she remains with the company to a maximum of four work weeks of paid vacation time. a. Let x represent the number of years she has worked for this employer and y represent the number of paid vacation days she has earned. Write an equation that models the relationship between these two variables. b. It has been five years since Lisa began working for this employer. How many paid vacation days has she earned? c. When will she reach the maximum number of paid vacation days allowed?

Short Answer

Expert verified
a. \(y = 3x + 2\)\nb. Lisa has earned 17 days of paid vacation after 5 years of work. \nc. Lisa would reach the maximum number of paid vacation days allowed in 6 years.

Step by step solution

01

Solving for Variable Equation

The first part of the problem requires determining the relationship between two variables. Given that the employer started with 2 days of paid vacation and adds 3 additional days for each year worked, we can form the linear equation in the form \(y = mx + c\). In this case, m (the slope) represents the additional days per year, which is 3. c (the interception) represents the initial days of paid vacation which is 2. So, the equation becomes \(y = 3x + 2\).
02

Calculation of Vacation Days for Specific Year

The second part requires to determine the number of paid vacation days Lisa has earned after working for 5 years. We need to substitute \(x = 5\) in our equation \((y = 3 * 5 + 2)\), which simplifies to \(y = 15 + 2 = 17\). So, Lisa has earned 17 days of paid vacation after 5 years of work.
03

Determine When Maximum Vacation Days Is Reached

The last part requires finding when the maximum allowable vacation time is reached. Given that the maximum work weeks of paid vacation time is 4 weeks and each week contains 5 workdays, max vacation days equals to \(4 * 5 = 20\) days. We set \(y = 20\) in our equation \((20 = 3x + 2)\) and then solve for \(x\). Simplifying gives \(x = (20 - 2) / 3 \) which results in \(x = 6\). This means that Lisa would reach the maximum number of paid vacation days allowed in 6 years.

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Key Concepts

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

Vacation Days Calculation
Predicting how many vacation days you can accumulate is exciting, especially when it involves understanding how your benefits grow over time. In Lisa's case, she began her job with 2 days of paid vacation. Her employer also promised 3 additional days each year. To find out how many vacation days Lisa could have at any given time, we use the formula:

  • Initial vacation days: 2
  • Additional vacation days each year: 3
To calculate Lisa's total vacation days over years, the formula used is:

\(y = 3x + 2\)

This equation helps determine the amount of vacation days based on the number of years \(x\). Using this simple equation, Lisa can see her potential future vacations build as she spends more time working at her company.
Modeling Relationships Between Variables
Modeling relationships between variables is essential for understanding how changes in one aspect of a scenario can impact another. In this problem, there are two main variables: the number of years \(x\) Lisa has worked and her total paid vacation days \(y\).

The linear equation \(y = 3x + 2\) represents this relationship. Here, \(x\) is an independent variable, representing the number of years, while \(y\) represents the dependent variable 鈥� the number of vacation days influenced by \(x\). The equation has two primary components:

  • Slope \(m = 3\): This shows how many additional vacation days Lisa earns each year.
  • Intercept \(c = 2\): This is the starting point, indicating the days given when she started the job.
This simple model helps forecast Lisa's vacation time with ease, as it directly links work tenure with vacation benefits.
Solving Linear Equations
Solving linear equations involves manipulating variables to find unknown values. For Lisa鈥檚 vacation scenario, the task is to identify at what point she reaches her maximum allowable vacation days, which is 20 days.

The equation to solve is:

\(20 = 3x + 2\)

Steps to solve for \(x\) include:
  • Subtract the intercept from both sides: \(20 - 2 = 3x\)
  • Resulting in: \(18 = 3x\)
  • Finally, divide by 3: \(x = 6\)
Through these steps, we discover that Lisa will hit her maximum of 20 vacation days after 6 years of working. This logical progression shows how understanding and solving linear equations is crucial, especially for planning future events like vacation time.

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Most popular questions from this chapter

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