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Chapter 3: Problem 6

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)?

Short Answer

Expert verified
Commercials watched for 5 hours: 75 minutes. Hours for 60 minutes of commercials: 4 hours. Function: \( C = 15T \).

Step by step solution

01

Set Up the Coordinate System

On the horizontal axis (x-axis), label the units as hours of television watched (T). On the vertical axis (y-axis), label the units as the minutes of commercials (C). Begin your graph with both axes starting at zero.
02

Identify the Initial Point

When no television is watched, no commercials are watched. This corresponds to the point (0,0) on your graph. Plot this initial point on the coordinate system.
03

Plot the One-Hour Point

For 1 hour of TV watched, there are 15 minutes of commercials. From the (0,0) point, move 1 unit to the right and 15 units up, reaching the point (1,15). Plot this point on your graph.
04

Plot Additional Points

Each additional hour watched corresponds to an additional 15 minutes of commercials. Plot the points (2,30), (3,45), (4,60), and (5,75) by moving 1 unit to the right and 15 units up each time.
05

Draw the Line

Draw a straight line through the points (0,0), (1,15), (2,30), (3,45), (4,60), and (5,75). This line represents the relationship between hours of TV watched and minutes of commercials.
06

Determine Commercials for 5 Hours

From the graph, locate where 5 hours of TV watched meets the line. The corresponding y-value on the vertical axis is 75 minutes of commercials.
07

Estimate from Commercials to TV Hours

To estimate how many hours of TV corresponds to 60 minutes of commercials, find 60 on the y-axis and locate the corresponding x-value on your line. This should correspond to 4 hours of TV watched.
08

Fill in the Table

Fill in the table with the number of minutes of commercials watched: For 0 hours - 0 minutes, 1 hour - 15 minutes, 2 hours - 30 minutes, 3 hours - 45 minutes, 4 hours - 60 minutes, and 5 hours - 75 minutes.
09

Express as a Function

The relationship between hours watched and commercials can be expressed as a function: \( C = 15T \). Using this formula, for 5 hours of TV, \( C = 15 \times 5 = 75 \) minutes of commercials. This matches the graph result from step 6.

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Key Concepts

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

Coordinate System
In the world of mathematics, a coordinate system allows us to visually represent relationships between two variables. Think of it as a map, where each point can tell you something about the connected values. In this exercise, we set up our coordinate system with two axes. The horizontal axis, often called the x-axis, represents the amount of television watched, labeled as \( T \) in hours. The vertical axis, or y-axis, shows the total amount of commercials/ads watched, represented as \( C \) in minutes. Both these axes start at zero, grounding our system at the origin point, which is (0,0). This setup helps us easily see and understand the relationship between our two variables by plotting points on the graph.
Independent and Dependent Variables
When dealing with linear functions, it is important to understand the independent and dependent variables. The independent variable is the variable you can control or choose freely, and its changes affect the dependent variable. In our scenario, the amount of television watched, \( T \), is the independent variable. This is on the horizontal axis.

The dependent variable, however, is influenced by changes in the independent variable. Here, the dependent variable is the amount of commercials/ads watched, \( C \). It resides on the vertical axis because its value depends on how much television is consumed. This relationship helps us predict values and understand how one variable affects the other.
Graphing Linear Equations
Graphing linear equations involves plotting points on a coordinate system to visualize the relationship between variables. In this exercise, we plot the relationship between television watched and commercials seen. Let's illustrate the process:
  • Identify key points: Start with the obvious point (0,0) 鈥� no TV means no commercials.
  • Calculate and plot: For 1 hour of TV, commercials run 15 minutes, giving point (1,15). Continue by adding an hour each time and 15 minutes of commercials (e.g., (2,30), (3,45), etc.).
  • Draw the line: Connect these points with a straight line, which represents our linear equation and shows a constant rate of commercials per hour of TV.
Through this process, we visualize how each extra hour of TV translates to more commercials, clarifying the relationship.
Slope-Intercept Form
The slope-intercept form of a linear equation is expressed as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In our exercise, the linear function is \( C = 15T \), showing that for each hour of television (\( T \)), we have 15 minutes of commercials (\( C \)).

The slope \( m \) defines the steepness of the line, and in this case, it's 15. This indicates a consistent increase of 15 commercials per hour. The y-intercept \( b \) is typically where the line crosses the y-axis. Since our equation doesn't have an additional number, it's technically \( 0 \), starting at the origin. This form makes predictions simple: just multiply the TV hours by 15 to find the commercials, a clear and straightforward way to express their relationship.

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Most popular questions from this chapter

When Jessica drives her car to a work-related conference, her employer reimburses her approximately 45 cents per mile to cover the cost of gas and the wear-and-tear on the vehicle. a) Using distance traveled d, in miles, as the independent variable and amount reimbursed A, in dollars, as the dependent variable, make a sketch of a coordinate system and label appropriately. Mark distance every 5 miles and amount reimbursed every \(\$ 0.45 .\) b) For traveling 0 miles, the reimbursement is 0 . This corresponds to the point \((0,0)\). Plot it on your coordinate system. c) For a trip that requires her to drive a total of 5 miles, she is reimbursed \(5(0.45)=\$ 2.25\). This corresponds to the point \((5, \$ 2.25)\). Plot it. d) For each 5 miles you go to the right, you must go up \(\$ 2.25\) and plot the point. Do this until you reach 20 miles. e) Keeping in mind that we are modeling this discrete situation continuously, draw a line through your data points. f) In March, Jessica attends a conference that is only 5 miles away. Counting round trip, she travels 10 total miles. Use the graph to determine how much she is reimbursed. g) In December, she attends a conference 10 miles away. How long is her trip in total? Use the graph to determine how much she will be reimbursed. h) For longer trips, such as 200 total miles, you will probably need to make a much larger graph. And what if she travels 400 miles? Or further? It is limitations such as these that make it useful to find an equation that describes what the graph shows. To find the equation, we start with a table that helps us to understand the relationship between the dependent and independent variables. Complete the table below. i) Use the table from part (h) to come up with an equation that relates \(\mathrm{d}\) and \(\mathrm{A}\). j) Now, use the equation to determine the reimbursement amounts for trips of 200 miles and 400 miles.

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.

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.

Temperature is typically measured in degrees Fahrenheit in the United States; but it is measured in degrees Celsius in many other countries. The relationship between Fahrenheit and Celsius is linear. Let's choose the measurement of degrees in Celsius to be our independent variable and the measurement of degrees in Fahrenheit to be our dependent variable. Water freezes at 0 degrees Celsius, which corresponds to 32 degrees Fahrenheit; and water boils at 100 degrees Celsius, which corresponds to 212 degrees Fahrenheit. We can plot this information as the two points (0,32) and (100,212). The relationship is linear, so have the following graph: a) Use the graph to approximate the equivalent Fahrenheit temperature for 48 degree Celsius. b) To determine the rate of change of Fahrenheit with respect to Celsius, we draw a right triangle with sides parallel to the axes that connects the two points we know... Side PR is 100 degrees long, representing an increase in 100 degrees Celsius. Side \(\mathrm{RQ}\) is 180 degrees, representing an increase in 180 degrees Fahrenheit. Find the rate of increase of Fahrenheit per Celsius. c) The following table shows some values of temperatures in Celsius and their corresponding Fahrenheit readings. Zero degrees Celsius corresponds to 32 degrees Fahrenheit. Our rate is 9 degrees Fahrenheit for every 5 degrees Celsius, or \(9 / 5\) of a degree Fahrenheit for every 1 degree Celsius. So, for 1 degree Celsius, we increase the Fahrenheit reading by \(9 / 5\) degree, getting \(32+9 / 5(1)\). For 2 degrees Celsius, we increase by two occurrences of \(9 / 5\) degree to get \(32+9 / 5(2)\). Fill in the missing entries, following the pattern. d) Use the table to form an equation that gives degrees Fahrenheit in terms of degrees Celsius.

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.
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