510 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions 39. Health Care Spending Out-of-pocket spending in the United States for health care increased between 2009 and 2017. The function ƒ1x2 = 955e0.0196t models annual per capita out-of-pocket expenditures, in dollars. In this model, t represents the year, where t = 0 corresponds to 2009. (Data from Centers for Medicare and Medicaid Services.) (a) Estimate per capita out-of-pocket spending on health care in 2017 to the nearest dollar. (b) In what year did per capita spending reach $1033? 40. Recreational Expenditures Personal consumption expenditures for recreation in billions of dollars in the United States during the years 2009 through 2017 can be approximated by the function A1t2 = 673e0.0369t, where t = 0 corresponds to the year 2009. Based on this model, how much were recreational expenditures in 2017 to the nearest billion dollars? (Data from U.S. Bureau of Economic Analysis.) 41. Housing and Utilities Costs Household consumption expenditures for housing and utilities in billions of dollars in the United States during the years 2012 through 2017 can be approximated by H= 1985e0.0421t, where t is the number of years since 2012. Find H to the nearest billion dollars for each year. (Data from U.S. Bureau of Economic Analysis.) (a) 2013 (b) 2015 (c) 2017 42. Evolution of Language The number of years, n, since two independently evolving languages split off from a common ancestral language is approximated by n ≈-7600 log r, where r is the proportion of words from the ancestral language common to both languages. Find each of the following to the nearest year. (a) Find n if r = 0.9. (b) Find n if r = 0.3. (c) How many years have elapsed since the split if half of the words of the ancestral language are common to both languages? 43. School District Growth Student enrollment in the Wentzville School District, one of the fastest-growing school districts in the state of Missouri, has projected growth as shown in the graph. 13–14 14–15 15–16 16–17 17–18 18–19 19–20 20–21 21–22 22–23 23–24 14,000 0 15,000 16,000 17,000 18,000 19,000 20,000 School Year Projected Enrollment Enrollment Projections Data from Wentzville School District. x y 14,225 19,915
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