/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 When Lisa started at her current... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Interpret the quote in the context of your experiences with money.

When George started his current job, his employer told him that at the end of the fi rst year, he would receive two vacation days. After each year worked, his number of vacation days would double up to fi ve work weeks of paid vacation. a. Let x represent the work year and y represent the number of paid vacation days. Write an equation that models the relationship between these two variables. b. How many vacation days will he have earned after four years? c. In what year will he have maxed out his vacation days?

Sean is paid biweekly. His annual salary is \(\$ 42,500 .\) What is his biweekly salary to the nearest cent?

Marty is working with a math problem that defies intuition. He is going to pay his gardener for the entire month of July. He will pay the gardener every day. On the first day, he will pay the gardener \(0.01. On the second day, he will pay double the first day, \)0.02. On the third day, he will double the second day鈥檚 pay and pay \(0.04. a. Make a grid that looks like a calendar with 7 columns and 5 rows. b. Fill in the dates from July 1鈥揓uly 31. c. Enter the amount Marty pays his gardener on each day. d. On what day will the gardener鈥檚 pay exceed \)1,000,000 for the first time? e. If x represents the day and y represents the salary for that day, draw a scatterplot for the fi rst two weeks of July.

Justin is a golf pro. He works eight months per year, and is paid \(76,000. During the winter months, he teaches golf privately and earns another \)12,500. What is his average monthly salary based on his yearly earnings?
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.