904 CHAPTER 9 Systems and Matrices 78. Planning a Diet In Exercise 77, suppose that, in addition to the conditions given there, foods A and B cost $0.02 per gram, food C costs $0.03 per gram, and a meal must cost $8. Is a solution possible? (Modeling) Age Distribution in the United States Use matrices to solve each problem. Let x = 0 represent 2018 and x = 35 represent 2053. Round values to four decimal places as necessary and percents to the nearest tenth. 79. In 2018, 16.0% of the population was 65 or older. By 2053, this percent is expected to be 22.3%. The percent of the population aged 25–39 in 2018 was 20.5%. That age group is expected to include 18.7% of the population in 2053. (Data from U.S. Census Bureau.) (a) Assuming these population changes are linear, use the data for the 65-or-older age group to write a linear equation. Then do the same for the 25–39 age group. (b) Solve the system of linear equations from part (a). In what year will the two age groups include the same percent of the population? What is that percent? (c) Does the answer to part (b) suggest that the number of people in the U.S. population aged 25–39 is decreasing? Explain. 80. In 2018, 19.3% of the U.S. population was aged 40 –54. This percent is expected to decrease to 18.3% in 2053. (Data from U.S. Census Bureau.) (a) Write a linear equation representing this population change. (b) Solve the system containing the equation from part (a) and the equation from Exercise 79 for the 65-or-older age group. Give the year in which these two age groups will include the same percent of the population. What is that percent? (Modeling) Solve each problem using matrices. 81. Athlete’s Weight and Height The relationship between a professional basketball player’s height H (in inches) and weight W (in pounds) was modeled using two different samples of players. The resulting equations that modeled the two samples were W= 7.46H- 374 and W= 7.93H- 405. (a) Use each equation to predict the weight of a 6 ft 11 in. professional basketball player to the nearest pound. (b) According to each model, what change in weight, to the nearest hundredth pound, is associated with a 1-in. increase in height? (c) Determine the weight and height, to the nearest unit, where the two models agree.
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