Homework Sourse83 The percentage p of the United States population that was
83.) The percentage, p, of the United States population that was foreign-born x years after 1920 can be modeled by the formula: p = 0.004x2 - 0.36x + 14
a.) According to the model, what percentage of the U.S. population was foreign born in 2000?
b.) If trends shown by the model continue, in which year will 18% of the U.S. population be foreign-born?
Solution
a) The percentage, p, of the United States population that was foreign-born x years after 1920 is given by the formula: p = 0.004x2 - 0.36x + 14. Now, 2000 is 79 years after 1920, so that x = 79. Then p = 0.004 ( 79)2 - 0.36 (79) + 14 = 0.004(6241) - 0.36 (79) + 14 = 24.964 - 28.44 + 14 = 10.524. Thus 10.524 % of the Unites States population was foreign born in the year 2000.
b) Let 18 % of the US population be foreign born x years after 1920. Then 18 = 0.004x2 - 0.36x + 14. On multiplying both the sides by 1000, we get 4x2 - 360x + 14000 = 18000 or, 4x2 - 360x - 4000 = 0 or, x2 - 90x - 1000 = 0 ( on dividing both the sides by 4) Then, x2 - 100x + 10x - 1000 = 0 or, x ( x -100) + 10 (x - 100) = 0 or, ( x - 100)( x + 10) = 0. Thus either x = 100 or, x = -10 i.e. 18 % of the United States population was foreign born either 10 years before 1920, i.e. in 1909 or 100 years after 1920, i.e. in 2021.
NOTE: There is an ambiguity here. If we assume that x years after 1920 will be 1920 + x and not 1920 + x + 1 as we have assumed, then in part a), the year 2000 will be 80 years after 1920 and the answer will be 0.004 ( 80)2 - 0.36 (80) + 14 = 0.004(6400) - 0.36 (80) + 14 = 25.6 - 28.8 + 14 = 10.8. For part b) , we will have the answer as either 1910 or 2020.
