{"id":6651,"date":"2020-10-03T16:09:34","date_gmt":"2020-10-03T16:09:34","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-beginalgebra\/?post_type=chapter&p=6651"},"modified":"2026-02-05T07:50:51","modified_gmt":"2026-02-05T07:50:51","slug":"3-5-interpreting-slope","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-elementaryalgebra\/chapter\/3-5-interpreting-slope\/","title":{"raw":"3.5: Applications of Slope","rendered":"3.5: Applications of Slope"},"content":{"raw":"
\r\n

section 3.5 Learning Objectives<\/h3>\r\n3.5: Applications of Slope<\/strong>\r\n
    \r\n \t
  • Find the average rate of change<\/li>\r\n \t
  • Use a linear equation to make a prediction<\/li>\r\n<\/ul>\r\n<\/div>\r\n \r\n\r\nIn the previous section, we learned how to find the slope of a line.\u00a0 In this section we will look at applications of slope and the corresponding linear functions.\r\n

    Rate and Unit Rate<\/h2>\r\nA\u00a0rate<\/strong> is a ratio of two quantities with different units. The presence of units implies that this is a concept used in many real world\u00a0 applications.\u00a0 Often, it is useful to simplify a rate to\u00a0a\u00a0unit rate<\/strong>, which has a denominator of 1.\r\n
    \r\n

    Example 1<\/h3>\r\nA.\u00a0\u00a0<\/strong>Suppose you drive 216 miles on 9 gallons of gas. Express the rate and the unit rate comparing miles traveled to gallons of gas used.\r\n\r\nB.\u00a0\u00a0<\/strong>If you instead drove 223 miles on 9 gallons of gas, compute the miles traveled per gallon of gas. Round your answer to the nearest tenth.\r\n\r\n[reveal-answer q=\"882437\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"882437\"]\r\n\r\nA<\/strong>.\u00a0\u00a0<\/strong>First, we must be careful with the order we use to set up our rate.\u00a0 Since we are told to compare miles to gallons, we will put the miles in the numerator and gallons in the denominator. To set up the rate, we simply use the given values.\r\n

    [latex]Rate=\\frac{216 \\hspace{.05in} miles}{9 \\hspace{.05in} gallons}[\/latex], or equivalently [latex]216[\/latex] miles per [latex]9[\/latex] gallons<\/p>\r\nTo transform this into a unit rate, we can simplify the fraction to lowest terms since 9 divides evenly into 216.\r\n

    [latex]Unit \\hspace{.05in} Rate=\\frac{24 \\hspace{.05in} miles}{1 \\hspace{.05in} gallon}[\/latex], or [latex]24 \\hspace{.05in} miles \\hspace{.05in} per \\hspace{.05in} gallon[\/latex]<\/p>\r\nB.\u00a0\u00a0<\/strong>Now, the rate is [latex]\\frac{223 \\hspace{.05in} miles}{9 \\hspace{.05in} gallons}[\/latex]. However, you may notice that 223 is not divisible by 9. In this case, we just do the division, [latex]223 \\div 9[\/latex], potentially with the aid of a calculator. Round to the nearest tenth produces\r\n

    [latex]Unit \\hspace{.05in} Rate=24.8 \\hspace{.05in}miles \\hspace{.05in} per \\hspace{.05in} gallon[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n \r\n

    Find the average rate of change<\/h2>\r\nA\u00a0rate of change<\/strong> is a rate comparing the relative changes in two related variables.\u00a0 \"Change\" is computing by finding the difference between two.\u00a0 Does a fraction comparing the differences in two variables<\/em> sound familiar?\u00a0 If you are thinking slope, you are correct!\r\n\r\nThe slope of a function that describes, real, measurable quantities is a rate of change.\u00a0 In these functions, the slope describes the change in one quantity per change in another quantity.\u00a0 We can use what we know about slope to find the rate of change.\r\n\r\nPreviously, we learned that Slope =\u00a0[latex]\\frac{\\text{Rise}}{\\text{Run}}[\/latex].<\/em>\u00a0We will use this definition to find the rate of change in the following example.\r\n\r\nA candle has a starting length of 10 inches.\u00a0 Thirty minutes after lighting it, the length is 7 inches.\u00a0 Determine the rate of change in the length of the candle as it burns.\r\n\r\n\"Line\r\n\r\nThe graph shows the candle length as a function of time.\u00a0 The horizontal axis is time in minutes and the vertical axis is the candle length in inches.\u00a0 Since the candle has a starting length of 10 inches, we can represent that on the graph by the point (0, 10).\u00a0 After 30 minutes, the candle is 7 inches.\u00a0 This can be represented by the point (30, 7).\u00a0 To find the rate of change, we need to determine the slope of the line.\r\n

    Rate of Change = Slope = [latex]\\frac{\\text{Rise}}{\\text{Run}}[\/latex]<\/p>\r\nLook at the points (0, 10) and (30, 7) on the graph.\u00a0 To move from (0, 10) to (30, 7) we would go down 3 units and to the right 30 units.\u00a0 Be sure to pay attention to the scale of each axis.\u00a0 Therefore, the Rise<\/em> would be -3.\u00a0 (It is negative because we went down instead of up.)\u00a0 The Run<\/em> would be 30.\r\n

    Rate of Change = Slope = [latex]\\frac{\\text{Rise}}{\\text{Run}}=\\frac{-3\\text{ inches}}{30 \\text{ minutes}}=-0.1[\/latex] inches per minute<\/p>\r\nSo, the candle length is decreasing by 0.1 inches each minute.\r\n\r\nWe could also find the rate of change using the two ordered pairs and the following definition of slope:\r\n

    Rate of Change = Slope= [latex]\\frac{y_2-y_1}{x_2-x_1}=\\frac{7\\text{ inches}-10\\text{ inches}}{30\\text{ minutes}- 0\\text{ minutes}}=\\frac{-3\\text{ inches}}{30\\text{ minutes}}=-0.1[\/latex] inches per minute.<\/p>\r\nIn this candle example, the scales on the x-axis and y-axis were different from each other. The y-axis was in units of 1, while the x-axis was in units of 10. If you'd like to see another example of how to find the slope (or rate of change) of a line with different scales on the axes, see the video below:\r\n\r\n[embed]https:\/\/www.youtube.com\/watch?v=JqeOivUbYGk[\/embed]\r\n

    \r\n

    Example 2<\/h3>\r\nGiven the information in the table below, what is the average rate of change in the number of living wage jobs from 1998 to 2000?\r\n\r\nLiving Wage Jobs\r\n\r\n\r\n\r\n\r\n
    Year<\/strong><\/td>\r\n1998<\/td>\r\n1999<\/td>\r\n2000<\/td>\r\n2001<\/td>\r\n<\/tr>\r\n
    Jobs<\/strong><\/td>\r\n685<\/td>\r\n722<\/td>\r\n760<\/td>\r\n798<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"230016\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"230016\"]\r\n\r\nFirst, we need to create two ordered pairs from the data.\r\n\r\n[latex](1998, 685)[\/latex] and\u00a0[latex](2000, 760)[\/latex]\r\n\r\nNotice that the \u201cyears\u201d are the x-coordinates and the \u201cnumber of jobs\u201d are the y-coordinates.\u00a0 That is so the number of jobs will be in the numerator in our slope formula.\u00a0 Since we were asked to find the rate of change in jobs per year, we need the number of jobs to be in the numerator and the years to be in the denominator.\r\n\r\nNext, we are going to use these two ordered pairs in the slope formula to find the average rate of change in jobs per year.\r\n\r\nRate of Change = Slope= [latex]\\frac{y_2-y_1}{x_2-x_1}[\/latex]\r\n\r\nRate of Change = Slope=[latex]\\frac{760-685}{2000-1998}[\/latex]\r\n\r\nRate of Change = Slope= [latex]\\frac{75}{2}[\/latex]\r\n\r\nRate of Change = Slope= 37.5 jobs\/year\r\n\r\nThe number of living wage jobs increased by 37.5 jobs per year.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n
    \r\n

    Example 3<\/h3>\r\nA city\u2019s population in 2012 was 3,267,100.\u00a0 In 2002, the population was 3,289,200.\u00a0 What is the average rate of change in the population per year from 2002 to 2012?\r\n\r\n[reveal-answer q=\"54127\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"54127\"]\r\n\r\nFirst, we need to create two ordered pairs from the data. It is important to label the variables correctly. When time is involved, time will usually be our x-variable. This is confirmed in that we want the rate of change in the population (so population on top) per year (therefore, years on bottom).\r\n

    [latex](2012, 3267100)[\/latex] and [latex](2002, 3289200)[\/latex]<\/p>\r\nUse these two ordered pairs in the slope formula to find the average rate of change in people per year.\r\n\r\nRate of Change = Slope= [latex]\\frac{y_2-y_1}{x_2-x_1}=\\frac{\\text{Change in population}}{\\text{Change in time}}[\/latex]\r\n\r\nRate of Change = Slope=[latex]\\frac{3289200-3267100}{2002-2012}[\/latex]\r\n\r\nRate of Change = Slope= [latex]\\frac{22100}{-10}[\/latex]\r\n\r\nRate of Change = Slope= [latex]\\frac{2210\\text{ people}}{1\\text{ year}}[\/latex]\r\n\r\nThe population decreased by an average of 2,210 people\/year.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

    \r\n

    Example 4<\/h3>\r\nOn a bicycle, Michelle rides for 3 hours and is 35 miles from her house.\u00a0 After riding for 10 hours, she is 112 miles away.\u00a0 What is Michelle\u2019s average rate in miles per hour during her trip?\r\n\r\n[reveal-answer q=\"189388\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"189388\"]\r\n\r\nFirst, we need to create two ordered pairs from the data.\u00a0 Again, it is important to label the variables correctly. When time is involved, time will usually be our x-variable. We will make two ordered pairs to represent (hours, miles).\r\n

    [latex](3, 35)[\/latex] and [latex](10, 112)[\/latex]<\/p>\r\nUse these two ordered pairs in the slope formula to find the average rate in miles per hour during her trip.\r\n\r\nRate of Change = Slope= [latex]\\frac{y_2-y_1}{x_2-x_1}=\\frac{\\text{Change in miles}}{\\text{Change in time}}[\/latex]\r\n\r\nRate of Change = Slope=[latex]\\frac{112-35}{10-3}[\/latex]\r\n\r\nRate of Change = Slope= [latex]\\frac{77}{7}[\/latex]\r\n\r\nRate of Change = Slope=\u00a0[latex]\\frac{11\\text{ miles}}{1\\text{ hour}}[\/latex]\r\n\r\nMichelle\u2019s average rate was 11 miles per hour (mph).\r\n\r\n \r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

    Use a linear model to make a prediction<\/h2>\r\nIn the first two examples, we will be given the equation and asked to make a prediction about the data using that equation.\r\n
    \r\n

    Example 5<\/h3>\r\nThe following linear model describes the change in median home values in Hawaii between 1950 and 2000.\r\n

    [latex]f(x)=3966x+74,400[\/latex]<\/p>\r\n

    where [latex]x=[\/latex] the number of years since 1950, and [latex]f(x)=[\/latex] the median value of a house.<\/p>\r\nUse this linear model to predict the median value of a house in Hawaii in the year 2022.\r\n\r\n[reveal-answer q=\"70338\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"70338\"]\r\n\r\nSince [latex]x[\/latex] represents the number of years since 1950, we will need to subtract 1950 from 2022 to determine what our\u00a0 [latex]x[\/latex]-value is.\r\n

    [latex]2022-1950=72[\/latex]<\/p>\r\nNext, we will substitute [latex]x=72[\/latex] into the function.\r\n

    [latex]f(x)=3966x+74,400[\/latex]<\/p>\r\n

    [latex]f(72)=3966(72)+74,400[\/latex]<\/p>\r\n

    [latex]f(72)=285,552+74,00[\/latex]<\/p>\r\n

    [latex]f(72)=359,952[\/latex]<\/p>\r\nTherefore, based on this linear model, we predict that the median value of a house in Hawaii in 2022 will be $359,952.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nSometimes when a linear model represents a set of data or observations, the y<\/em>-intercept can be interpreted as a starting point.\u00a0 Let\u2019s look at the previous example and determine what the y-intercept represents.\r\n\r\nThe y<\/em>-intercept of any equation is the point where the x<\/em>-coordinate is 0.\u00a0 Substituting \u00a0into the linear model give us the following:\r\n

    [latex]f(x)=3966x+74,400[\/latex]<\/p>\r\n

    [latex]f(0)=3966(0)+74,400[\/latex]<\/p>\r\n

    [latex]f(0)=74,400[\/latex]<\/p>\r\nWhat does this result mean?\u00a0 Since [latex]x[\/latex] represents the number of years since 1950, [latex]x=0[\/latex] would represent the year 1950.\u00a0 Therefore, [latex]f(0)=74,400[\/latex] means that in 1950 the median value of a house in Hawaii was $74,400.\r\n\r\nThe following video shows another example of how to make a prediction with the home value data.\r\n\r\nhttps:\/\/youtu.be\/Bw9XjDAl-K0\r\n

    \r\n

    Example 6<\/h3>\r\nData on tuition and mid-career salary were collected from a number of universities and colleges.\u00a0 The result of the data collection is the following linear model:\r\n

    [latex]f(x)=-0.97x+159,000[\/latex]<\/p>\r\n

    where [latex]x=[\/latex] annual tuition, and [latex]f(x)=[\/latex] average mid-career salary of graduates.<\/p>\r\n

    A.\u00a0 What is the slope of this linear model?<\/p>\r\n

    B.\u00a0 According to this model, what is the average salary for a graduate of a college or university where the annual tuition is $30,000?<\/p>\r\n

    C.\u00a0 According to this model, what happens to mid-career salary as tuition increases?<\/p>\r\n[reveal-answer q=\"526875\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"526875\"]\r\n

    A.\u00a0 The slope is [latex]m=-0.97[\/latex].<\/p>\r\n

    B.\u00a0 Since [latex]x[\/latex] represents the annual tuition, we will substitute 30,000 for [latex]x[\/latex]; or in other words, we are looking for [latex]f(30,000).<\/p>\r\n

    [latex]f(x)=-0.97x+159,000[\/latex]<\/p>\r\n

    [latex]f(30,000)=-0.97(30,000)+159,000[\/latex]<\/p>\r\n

    [latex]f(30,000)=-29,100+159,000[\/latex]<\/p>\r\n

    [latex]f(30,000)=129,900[\/latex]<\/p>\r\n

    Therefore, based on this model, a graduate of a college or university where the annual tuition is $30,000 can expect an average mid-career salary of $129,900.<\/p>\r\n

    C.\u00a0 We already identified the slope to be [latex]m=-0.97[\/latex], but what does this mean in the context of the problem? Recall from the first half of this section that slope is the rate of change, specifically the change in [latex]y[\/latex] relative to changes in [latex]x[\/latex]. In this problem, [latex]x[\/latex] represents tuition while [latex]y[\/latex] represent mid-career salary.<\/p>\r\n

    Based on the negative slope, we conclude that mid-career salary decreases as tuition increases.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

    \r\n

    Example 7<\/h3>\r\nThe cost to produce Mario\u2019s Skyrockets is given by the following linear model:\r\n

    [latex]C(x)=25x+840[\/latex]<\/p>\r\n

    where [latex]x=[\/latex] the number of skyrockets produced, and [latex]C(x)=[\/latex] the total cost.<\/p>\r\nUse this linear model to predict:\r\n

    A. What is the slope?<\/p>\r\n

    B.\u00a0 What is the cost to produce 20 skyrockets?<\/p>\r\n

    C.\u00a0 According to this model, what happens to cost as the number of rockets produced increases?<\/p>\r\n[reveal-answer q=\"793635\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"793635\"]\r\n

    A.\u00a0 The slope is [latex]m=25[\/latex].<\/p>\r\n

    B.\u00a0 Since [latex]x=[\/latex] the number of skyrockets produced, we will substitute [latex]x=20[\/latex] into the function.<\/p>\r\n

    [latex]C(x)=25x+840[\/latex]<\/p>\r\n

    [latex]C(20)=25(20)+840[\/latex]<\/p>\r\n

    [latex]C(20)=500+840[\/latex]<\/p>\r\n

    [latex]C(20)=1340[\/latex]<\/p>\r\n

    Therefore, the cost to produce 20 skyrockets is $1,340.<\/p>\r\n

    C. In context, the slope here represent the average rate of change in cost relative to the change in rockets produced. So, the slope of [latex]m=25[\/latex], or equivalently, [latex]m=\\frac{25}{1}[\/latex] indicates that the cost increases by $25 for each additional rocket produced.\u00a0 Therefore, cost increases as the number of rockets produced increases.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video gives another example where we must solve for the\u00a0x<\/em>-value.\r\n\r\nhttps:\/\/youtu.be\/5W0qq8saxO0\r\n

    Summary<\/h2>\r\nIn this section we looked at a couple of applications of slope and linear models.\u00a0 We learned that in a real world context, slope represents the average rate of change. This leads to linear functions that can be used to model such real world situations. These models can then be used to make predictions for given inputs.","rendered":"
    \n

    section 3.5 Learning Objectives<\/h3>\n

    3.5: Applications of Slope<\/strong><\/p>\n

      \n
    • Find the average rate of change<\/li>\n
    • Use a linear equation to make a prediction<\/li>\n<\/ul>\n<\/div>\n

       <\/p>\n

      In the previous section, we learned how to find the slope of a line.\u00a0 In this section we will look at applications of slope and the corresponding linear functions.<\/p>\n

      Rate and Unit Rate<\/h2>\n

      A\u00a0rate<\/strong> is a ratio of two quantities with different units. The presence of units implies that this is a concept used in many real world\u00a0 applications.\u00a0 Often, it is useful to simplify a rate to\u00a0a\u00a0unit rate<\/strong>, which has a denominator of 1.<\/p>\n

      \n

      Example 1<\/h3>\n

      A.\u00a0\u00a0<\/strong>Suppose you drive 216 miles on 9 gallons of gas. Express the rate and the unit rate comparing miles traveled to gallons of gas used.<\/p>\n

      B.\u00a0\u00a0<\/strong>If you instead drove 223 miles on 9 gallons of gas, compute the miles traveled per gallon of gas. Round your answer to the nearest tenth.<\/p>\n

      Show Solution<\/span><\/p>\n
      \n

      A<\/strong>.\u00a0\u00a0<\/strong>First, we must be careful with the order we use to set up our rate.\u00a0 Since we are told to compare miles to gallons, we will put the miles in the numerator and gallons in the denominator. To set up the rate, we simply use the given values.<\/p>\n

      [latex]Rate=\\frac{216 \\hspace{.05in} miles}{9 \\hspace{.05in} gallons}[\/latex], or equivalently [latex]216[\/latex] miles per [latex]9[\/latex] gallons<\/p>\n

      To transform this into a unit rate, we can simplify the fraction to lowest terms since 9 divides evenly into 216.<\/p>\n

      [latex]Unit \\hspace{.05in} Rate=\\frac{24 \\hspace{.05in} miles}{1 \\hspace{.05in} gallon}[\/latex], or [latex]24 \\hspace{.05in} miles \\hspace{.05in} per \\hspace{.05in} gallon[\/latex]<\/p>\n

      B.\u00a0\u00a0<\/strong>Now, the rate is [latex]\\frac{223 \\hspace{.05in} miles}{9 \\hspace{.05in} gallons}[\/latex]. However, you may notice that 223 is not divisible by 9. In this case, we just do the division, [latex]223 \\div 9[\/latex], potentially with the aid of a calculator. Round to the nearest tenth produces<\/p>\n

      [latex]Unit \\hspace{.05in} Rate=24.8 \\hspace{.05in}miles \\hspace{.05in} per \\hspace{.05in} gallon[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

       <\/p>\n

      Find the average rate of change<\/h2>\n

      A\u00a0rate of change<\/strong> is a rate comparing the relative changes in two related variables.\u00a0 “Change” is computing by finding the difference between two.\u00a0 Does a fraction comparing the differences in two variables<\/em> sound familiar?\u00a0 If you are thinking slope, you are correct!<\/p>\n

      The slope of a function that describes, real, measurable quantities is a rate of change.\u00a0 In these functions, the slope describes the change in one quantity per change in another quantity.\u00a0 We can use what we know about slope to find the rate of change.<\/p>\n

      Previously, we learned that Slope =\u00a0[latex]\\frac{\\text{Rise}}{\\text{Run}}[\/latex].<\/em>\u00a0We will use this definition to find the rate of change in the following example.<\/p>\n

      A candle has a starting length of 10 inches.\u00a0 Thirty minutes after lighting it, the length is 7 inches.\u00a0 Determine the rate of change in the length of the candle as it burns.<\/p>\n

      \"Line<\/p>\n

      The graph shows the candle length as a function of time.\u00a0 The horizontal axis is time in minutes and the vertical axis is the candle length in inches.\u00a0 Since the candle has a starting length of 10 inches, we can represent that on the graph by the point (0, 10).\u00a0 After 30 minutes, the candle is 7 inches.\u00a0 This can be represented by the point (30, 7).\u00a0 To find the rate of change, we need to determine the slope of the line.<\/p>\n

      Rate of Change = Slope = [latex]\\frac{\\text{Rise}}{\\text{Run}}[\/latex]<\/p>\n

      Look at the points (0, 10) and (30, 7) on the graph.\u00a0 To move from (0, 10) to (30, 7) we would go down 3 units and to the right 30 units.\u00a0 Be sure to pay attention to the scale of each axis.\u00a0 Therefore, the Rise<\/em> would be -3.\u00a0 (It is negative because we went down instead of up.)\u00a0 The Run<\/em> would be 30.<\/p>\n

      Rate of Change = Slope = [latex]\\frac{\\text{Rise}}{\\text{Run}}=\\frac{-3\\text{ inches}}{30 \\text{ minutes}}=-0.1[\/latex] inches per minute<\/p>\n

      So, the candle length is decreasing by 0.1 inches each minute.<\/p>\n

      We could also find the rate of change using the two ordered pairs and the following definition of slope:<\/p>\n

      Rate of Change = Slope= [latex]\\frac{y_2-y_1}{x_2-x_1}=\\frac{7\\text{ inches}-10\\text{ inches}}{30\\text{ minutes}- 0\\text{ minutes}}=\\frac{-3\\text{ inches}}{30\\text{ minutes}}=-0.1[\/latex] inches per minute.<\/p>\n

      In this candle example, the scales on the x-axis and y-axis were different from each other. The y-axis was in units of 1, while the x-axis was in units of 10. If you’d like to see another example of how to find the slope (or rate of change) of a line with different scales on the axes, see the video below:<\/p>\n