The table shows the total personal income in a country (in billions ofdollars) for selected years from 1960 and projected to 2024. Completeparts (a) and (b) below.….Year196019701980y=(x²+x+(Type integers or decimals rounded to three decimal places as needed.)19902000Income($ billions)Year411.72008838.620142307.9 20184878.7 2024 22685.58429.9Income($ billions)12100.814728.619129.7a. These data can be modeled by a quadratic function. Write the equation of this function, with x equal to thenumber of years after 1960.b. In what year does the model predict the total personal income will reach $24.772 trillion?The unrounded model predicts the total personal income will reach $24.772 trillion in

When we want to predict the value of one variable, say y, from the given value of another variable, say x, we use Regression Equations of y on x given by, Using the method of least squares the values of a and b are determined by solving the following two normal equations:From the above normal equations, we find the values of a, b and c by solving the normal equations. Year x = Year - 1960 Income ($ billions) y x2 x3 x4 xy x2y 1960 0 411.7 0 0 0 0 0 1970 10 838.6 100 1000 10000 8386 83860 1980 20 2307.9 400 8000 160000 46158 923160 1990 30 4878.7 900 27000 810000 146361 4390830 2000 40 8429.9 1600 64000 2560000 337196 13487840 2008 48 12100.8 2304 110592 5308416 580838.4 27880243.2 2014 54 14728.6 2916 157464 8503056 795344.4 42948597.6 2018 58 19129.7 3364 195112 11316496 1109523 64352310.8 2024 64 22685.5 4096 262144 16777216 1451872 92919808 Total 324 85511.4 15680 825312 45445184 4475678 246986649.6 The normal equations are By solving we get, (rounded to 3 decimal places) Answer: The blanks will be filled up with 6.116, -49.282 and 620.418 respectively.To find the year when the total personal income will reach $24.772 trillion i.e. $24772 billion, we put y=24772 in the above equation and get, 24772=6.116x2-49.282x+620.418⇒6.116x2-49.282x+620.418-24772=0⇒6.116x2-49.282x-24151.582=0⇒x=49.282±-49.2822-4×6.116×-24151.5822×6.116=67, -59 Clearly, the value of x cannot be negative as it will imply earlier than 1960. So, the only value of x is 67. The corresponding year is: 1960+67=2027 Answer: The blank will be filled up with 2027.