Quadratic & Power Modelling

  1. Total personal income (TPI), is a measure of the sum-total income received by individuals from all sources. The BEA (Bureau of Economic Analysis) reports historical data for total personal income (in trillions of dollars) in the US according to this table:
    Years since 1970 10 15 20 25 30 35 40 45 50
    TPI (in trillions) 2.32 3.52 4.91 6.28 8.62 10.55 12.56 15.47 19.63
    1. Use technology find a function of the form \(g(t) = at^b\) that models TPI as a function of \(t\) years since 1970. Write \(g(t)\) down here with parameters rounded to three decimal places.
    2. What is the value of \(g(45)?\) What does this value represent within the context of the model? By what percent does this value differ from the figure reported by the BEA?
    3. Within the context of the model, what do the values of the constants \(a\) and \(b\) represent?
    4. What does the model predict total personal income in the US to be next year?
    5. By which year does the model predict the US will have a total personal income of $25 trillion?

  2. Here is select historical data from the UN estimating the population of the earth in billions.

    Year 1985 1990 1995 2000 2005 2010 2015 2020
    Population 4.86 5.32 5.74 6.15 6.56 6.99 7.43 7.84

    1. Use technology to perform regression to find a quadratic function that models the earth’s population (in billions) as a function of \(t\) years since 1980.

    2. According to your model, what is the current population of the earth?
    3. According to your model, what was the population of the earth in 1980?
    4. What are the \(x\)- and \(y\)-coordinates of the vertex of the graph of your function? What do these coordinates mean within the context of the model?
    5. According to this model, the population of the earth will hit a maximum and then begin to decline. What is the range of years during which the population will be declining?
    6. At some time since 1980 the earth’s population was exactly 5 billion people. Find the year during which the model indicates this happened. Then since the model indicates the earth’s population will eventually decline to zero, it must become exactly 5 billion again in the future; calculate the year this happens too.

  3. The frequency at which Hollywood is producing Batman movies is increasing dramatically. Here is data correlating the rank of each Batman movie (the 1st movie, 2nd movie, 3rd movie, etc) since the original 1943 film Batman starring Lewis Wilson, with the year it was released since 1900. (Excluding DC universe films that feature Batman but are not about Batman.)

    Movie Number 1 2 3 4 5 6 7 8 9 10 11 12 13
    Year Since 1900 43 49 66 82 92 95 97 105 108 112 116 117 122
    1. Use technology find a function of the form \(f(x) = ax^b\) that models the release date of a Batman movie given its rank.
    2. According to your model, what year should we expect the 30th Batman movie?
    3. According to this model, how many Batman movies should we expect to exist by 2080?
    4. According to this model, by what year should we start to expect that multiple Batman movies will be released per year, each and every year into the future?