117 1.2 Applications and Modeling with Linear Equations EXAMPLE 5 Modeling Prevention of Indoor Pollutants If a vented range hood removes contaminants such as carbon monoxide and nitrogen dioxide from the air at a rate of F liters of air per second, then the percent P of contaminants that are also removed from the surrounding air can be modeled by the linear equation P = 1.06F + 7.18, where 10 … F … 75. What flow F (to the nearest hundredth) must a range hood have to remove 50% of the contaminants from the air? (Data from Proceedings of the Third International Conference on Indoor Air Quality and Climate.) SOLUTION Replace P with 50 in the linear model, and solve for F. P = 1.06F + 7.18 Given model 50 = 1.06F + 7.18 Let P = 50. 42.82 = 1.06F Subtract 7.18. F ≈40.40 Divide by 1.06. To remove 50% of the contaminants, the flow rate must be 40.40 L of air per second. S Now Try Exercise 41. EXAMPLE 6 Modeling Health Care Costs The projected per capita health care expenditures in the United States, where y is in dollars, and x is years after 2010, are given by the following linear equation. y = 416x + 8042 Linear model (Data from Centers for Medicare and Medicaid Services.) (a) What were the per capita health care expenditures in the year 2018? (b) If this model continues to describe health care expenditures, when will the per capita expenditures reach $13,000? SOLUTION In part (a) we are given information to determine a value for x and asked to find the corresponding value of y, whereas in part (b) we are given a value for y and asked to find the corresponding value of x. (a) The year 2018 is 8 yr after the year 2010. Let x = 8 and find the value of y. y = 416x + 8042 Given model y = 416182 + 8042 Let x = 8. y = 11,370 Multiply and then add. In 2018, the estimated per capita health care expenditures were $11,370. (b) Let y = 13,000 in the given model, and find the value of x. y = 416x + 8042 Given model 13,000 = 416x + 8042 Let y = 13,000. 4958 = 416x Subtract 8042. x ≈11.9 Divide by 416. The x-value of 11.9 indicates that per capita health care expenditures are projected to reach $13,000 during the 11th year after 2010—that is, in 2021. S Now Try Exercise 45. 11 corresponds to 2010 + 11 = 2021.
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