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Chapter 3: Problem 4

Joe owes \(\$ 24,000\) in student loans. He has finished college and is now working. He can afford to pay \(\$ 1500\) per month toward his loans. a) Choose time in months as your independent variable and amount owed, in \$, as the dependent variable. On a sheet of graph paper, make a sketch of the coordinate system, using tick marks and labeling the axes appropriately. b) At time \(\mathrm{t}=0\), Joe has not yet paid anything toward his loans. To what point does this correspond? Plot this point on your coordinate system. c) After one month, he pays \(\$ 1500\). Beginning at the previous point, move 1 month to the right and \(\$ 1500\) down (down because the debt is decreasing). Plot this point. What are its coordinates? d) Each time you go 1 month to the right, you must move \(\$ 1500\) down. Continue doing this until his loans have been paid off. e) Keeping in mind that we are modeling this discrete situation continuously, draw a line through your data points. f) Use the graph to determine how many months it will take him to pay off the full amount of his loans.

Short Answer

Expert verified
It will take Joe 16 months to pay off his \( \$24,000 \) loan.

Step by step solution

01

Set Up the Coordinate System

Set up a coordinate system where the x-axis represents time in months and the y-axis represents the amount owed in dollars. Label the x-axis as 'Time (months)' and the y-axis as 'Amount Owed ($)'. Set appropriate tick marks for the range of months and amounts he owes.
02

Plot the Initial Point

At time \( t = 0 \), Joe has not paid anything, and he owes \( \$24,000 \). Thus, the point is \((0, 24000)\). Plot this point on the coordinate system.
03

Move to One-Month Payment Point

After one month, Joe pays \( \\(1500\), decreasing the amount owed to \( \\)22,500 \). Move 1 month to the right and \( \$1500 \) down from \((0, 24000)\) to reach the point \((1, 22500)\). Plot this point on the graph.
04

Continue Plotting Points Each Month

Continue the process: each month, move one unit to the right on the x-axis and \( \$1500 \) down on the y-axis until the amount owed reaches zero. For example, \( (2, 21000), (3, 19500), (4, 18000), \) etc., until \( y = 0 \).
05

Draw a Line Through the Points

Draw a straight line through all the plotted points, illustrating a linear relationship as the debt is paid monthly at a constant rate.
06

Determine the Months Required to Pay Off Debt

To find how many months it will take for Joe to pay off \( \\(24,000 \), use the linear function approach. He pays \( \\)1500 \) per month, so solve \( 24000 - 1500t = 0 \) for \( t \). Therefore, \( t = \frac{24000}{1500} \), which equals 16 months.

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Key Concepts

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

Linear Equations
In the world of mathematics, linear equations are our basic tools for dealing with straight-line scenarios. A linear equation is an algebraic equation that represents a straight line graphically. In Joe's case, the linear equation models how his debt decreases over time. This equation typically takes the form \( y = mx + b \), where \( y \) is the dependent variable, \( m \) is the slope (indicating the rate of change), \( x \) is the independent variable, and \( b \) is the y-intercept (the starting point).

For Joe, the decrease in student loan debt is a straight line; this is because he pays a consistent amount monthly. Thus, the dependent variable (amount owed) decreases steadily over time (independent variable). Here, each month represents an increase along the x-axis, and a payment of $1500 represents a decrease along the y-axis.

The linear equation representing Joe's situation is \( y = 24000 - 1500x \). Here, \( 24000 \) is the starting debt, and \( -1500 \) is the slope indicating the monthly payment. This equation helps us understand how much Joe owes at any time \( x \).
Coordinate System
The coordinate system is a mathematical grid that helps visualize relationships between two varying quantities. It's like a map for plotting data points and showcasing trends, as we have done with Joe's loan repayment.

For Joe's exercise, the x-axis represents time in months, while the y-axis represents the amount Joe owes in dollars. To set up the system, start with the x-axis labeled "Time (months)". Place tick marks to represent each month. Similarly, label the y-axis "Amount Owed (") and mark tick lines for every $1500.

This setup allows you to easily plot and track each payment's impact on the total debt. Each point on the graph corresponds to a specific month and the debt balance at that time, helping visualize Joe's progress in debt repayment. Seeing how the plotted line trends downward communicates visually what a table of numbers might fail to convey.
Debt Management
Debt management is crucial for financial health and involves strategies to repay debt effectively, ensuring payments are manageable over time. Joe's strategy involves fixed payments that will consistently chip away at his debt, making his plan predictable and stress-free.

Understanding Joe's plan is straightforward: he owes $24,000 and pays $1500 monthly. This approach offers several benefits:
  • Consistency: By paying a constant amount each month, Joe can predict when his debt will be fully paid.
  • Budgeting: Knowing he will be debt-free in 16 months allows Joe to plan his future finances.
  • Motivation: Seeing steady progress as the loan balance decreases each month can keep Joe motivated, ensuring he sticks to his repayment plan.
Properly managing debt with such clear steps aids in reducing financial stress and achieving financial freedom.
Graphing Skills
Graphing skills are fundamental in mathematics and everyday scenarios like Joe's. They allow for the visual representation of mathematical data or relationships, making complex information more digestible.

Plotting Joe's debt repayment accurately involves several steps:
  • Determine the axes: X-axis for time and y-axis for amount owed.
  • Place initial point: At \( t = 0 \), the amount owed is \(24,000, so point \((0, 24000)\) is marked.
  • Subsequent points: For every month \( t \), plot a new point by moving right one unit and down by \)1500, resulting in points like \((1, 22500)\), \((2, 21000)\), etc.
  • Draw a line: Connect all the points to form a straight line, showing a constant reduction in debt.
These graphing skills let Joe (and you) see a complete picture of the loan repayment process, making it clear and straightforward.

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Most popular questions from this chapter

On network television, a typical hour of programming contains 15 minutes of commercials and advertisements and 45 minutes of the program itself. a) Choose amount of television watched as your independent variable and place it on the horizontal axis. Let T represent the amount of television watched, in hours. Choose total amount of commercials/ads watched as your dependent variable and place it on the vertical axis. Let \(\mathrm{C}\) represent the total amount of commercials/ads watched, in minutes. Using a sheet of graph paper, make a sketch of a coordinate system and label appropriately. b) For 0 hours of programming watched, 0 minutes of commercials have been watched. To what point does this correspond? Plot it on your coordinate system. c) After watching 1 hour of program-ming, 15 minutes of commercials/ads have been watched. Beginning at the previous point, move 1 hour to the right and 15 minutes up. Plot this point. What are its coordinates? d) Each time you go 1 hour to the right, you must move 15 minutes up and plot a point. Continue doing this until you reach 5 hours of programming. e) Draw a line through your data points. f) Billy watches TV for five hours on Monday. Use the graph to determine how many minutes of commercials he has watched during this time. g) Suppose a person has watched one hour of commercials/ads. Use the graph to estimate how many hours of television he watched. h) The following table shows numbers of hours of programming watched as it relates to number of minutes of commercials/ads watched. For 0 hours of TV, 0 minutes of commercials/ads are watched. For each hour of TV watched, we must count 15 minutes of commercials/ads. So, for 1 hour, \(0+15(1)\) minutes of commercials are watched. For 2 hours, \(0+15(2)\) minutes; and so on. Fill in the missing entries. i) Express the amount of commercials/ads watched, C, as a function of the amount of television watched T. Use your equation to predict the amount of commercials/ads watched for 5 hours of television programming. Does this answer agree with your estimate from part (f)?

Earl the squirrel has only ten more days until hibernation. He needs to save 50 more acorns. He is tired of collecting acorns and so he is only able to gather 8 acorns every 2 days. a) Let t represent time in days and make it your independent variable. Let \(\mathrm{N}\) represent the number of acorns collected and make it your dependent variable. Set up an appropriately scaled coordinate system on a sheet of graph paper. b) At time \(\mathrm{t}=0\), Earl has collected zero of the acorns he needs. To what point does this correspond? Plot this point on your coordinate system. c) After two days ( \(\mathrm{t}=2\) ), Earl has collected 8 acorns. Beginning at the previous point, move 2 days to the right and 8 acorns up. Plot this point. What are its coordinates? d) Each time you go 2 days to the right, you must move 8 acorns up and plot a point. Continue doing this until you reach 14 days. e) Keeping in mind that we are modeling this discrete situation continuously, draw a line through your data points. f) Use the graph to determine how many acorns he will have collected after 10 days. Will Earl have collected enough acorns for his winter hibernation? g) Notice that the number of acorns collected is increasing at a rate of 8 acorns every 2 days. Reduce this to a rate that tells the average number of acorns that is collected each day. h) The table below lists the number of acorns Earl will have collected at various times. Some of the entries have been completed for you. For example, at \(\mathrm{t}=0\), Earl has no acorns, so \(\mathrm{N}=\) 0\. After one day, the amount increases by 4 , so \(\mathrm{N}=0+4\) (1). After two days, two increases have occurred, so \(\mathrm{N}=0+4(2)\). The pattern continues. Fill in the missing entries. i) Express the number of acorns collected, N, as a function of the time t, in days. j) Use your function to predict the number of acorns that Earl will have after 10 days. Does this answer agree with your estimate from part (f)?

Jodiah is saving his money to buy a Playstation 3 gaming system. He estimates that he will need \(950 to buy the unit itself, accessories, and a few games. He has \)600 saved right now, and he can reasonably put \(60 into his savings at the end of each month. Since the amount of money saved depends on how many months have passed, choose time, in months, as your independent variable and place it on the horizontal axis. Let t represent the number of months passed, and make a mark for every month. Choose money saved, in dollars, as your dependent variable and place it on the vertical axis. Let A represent the amount saved in dollars. Since Jodiah saves \)60 each month, it will be convenient to let each box represent \(60. Copy the following coordinate system onto a sheet of graph paper. a) At month 0, Jodiah has \)600 saved. This corresponds to the point (0, 600). Plot this point on your coordinate system. b) For the next month, he saved \(60 more. Beginning at point (0, 600), move 1 month to the right and \)60 up and plot a new data point. What are the coordinates of this point? c) Each time you go right 1 month, you must go up by $60 and plot a new data point. Repeat this process until you reach the edge of the coordinate system. d) Keeping in mind that we are modeling this discrete situation continuously, draw a line through your data points. e) Use your graph to estimate how much money Jodiah will have saved after 7 months. f) Using your graph, estimate how many months it will take him to have saved up enough money to buy his gaming system, accessories, and games.

According to NATO (the National Association of Theatre Owners), the average price of a movie ticket was \(5.65\) dollars in the year 2001. Since then, the average price has been rising each year by about \(20 \backslash\) cents. a) Choose year, beginning with 2000 , as the independent variable and make marks every year on the axis. Choose average ticket price, in dollars, as your dependent variable and begin at \(5.65\) dollars, with marks every \(10 \backslash\) cents above. Make a sketch of a coordinate system and label appropriately. b) In 2001, the average ticket price was \(5.65\) dollars, corresponding to the point (2001, 5.65). Plot it on your coordinate system. c) In 2002, one year later, the average price rose by about \(20 \backslash\) cents. Beginning at the previous point, move right by 1 year and up by \(20 \backslash\) cents and plot the point. What are its coordinates? d) Each time you go 1 year to the right, you must move up by \(20 \backslash\) cents and plot a point. Continue doing this until the year 2010 . e) Keeping in mind that we are modeling this discrete situation continuously, draw a line through your data points. f) Use the graph to estimate what year the average price of a ticket will pass \(7.00\) dollars.

A boat is \(200 \mathrm{ft}\) from a buoy at sea. It approaches the buoy at an average speed of \(15 \mathrm{ft} / \mathrm{s}\). a) Choosing time, in seconds, as your independent variable and distance from the buoy, in feet, as your dependent variable, make a graph of a coordinate system on a sheet of graph paper showing the axes and units. Use tick marks to identify your scales. b) At time \(\mathrm{t}=0\), the boat is \(200 \mathrm{ft}\) from the buoy. To what point does this correspond? Plot this point on your coordinate system. c) After 1 second, the boat has drawn \(15 \mathrm{ft}\) closer to the buoy. Beginning at the previous point, move 1 second to the right and \(15 \mathrm{ft}\) down (since the distance is decreasing) and plot a new data point. What are the coordinates of this point? d) Each time you go right 1 second, you must go down by \(15 \mathrm{ft}\) and plot a new data point. Repeat this process until you reach 12 seconds. e) Draw a line through your data points. f) When the boat is within 50 feet of the buoy, the driver wants to begin to slow down. Use your graph to estimate how soon the boat will be within 50 feet of the buoy.
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