Total personal income of the country in billions of dollars

Total personal income of the country (in billions of dollars) for selected years from 1957 to 2002 is given in the table. (a) These data can be modeled by an exponential function. Write the equation of this function, with x as the number of/ears after 1957. (b) If this model is accurate, what will be the country\'s total personal income in 2007? (c) In what year does the model predict the total personal income will reach $23 trillion? (a) The equation of an exponential function that models the data is y =.

Solution

a) let x be number of years after 1957 ,

let the general form of exponential function be

y = a + b * c^x

points ( 0,409.1),(10,840.6),(20,2307.3),(30,4877.9),(40,8422.7),(45,10238.6) passes through it

taking ( 0,409.1),(20,2307.3),(40,8422.7) into consideration for calculating constant a,b,c ,we get

a + b = 409.1

a+b*c²⁰ = 2307.3

a + b*c⁴⁰ = 8422.9

we get a = -444.257,b = 854.357 ,c = 1.06024

so

y = -444.257 + 854.357*1.060^x

b) personal income in 2007 will be ,

when x = 50

y = 15300 (approx)

c) year in which personal income will reach 23 trillion will be

x = 412 , so in the year 2369