Question 1190019
the function is h(x) = 23.4 * 1.08 ^ x
x = 5 corresponds to the year 1995.
therefore, x = 0 corresponds to the year 1990, because 1995 - 1990 = 5.
you have:
1990 = 0
1991 = 1
1992 = 2
1993 = 3
1994 = 4
1995 = 5.
the value of x is equal to 23.4 because 23.4 * 1.08 ^ 0 = 23.4.
the value of the investment in 2011 is equal to 23.4 * 1.08 ^ (2011 - 1990) = 23.4 * 1.08 ^ 21 = 117.7917089.
to find the first year that the expenditures exceeded 105 billion, replace h(x) with 105 to get:
105 = 23.4 * 1.08 ^ x
divide both sides of the equation by 24.4 to get:
105/23.4 = 1.08 ^ x
take the log of both sides of the equation to get:
log(105/23.4) = log(1.08 ^ x)
this becomes log(105/23.4) = x * log(1.08) by the laws of log exponentiation
divide both sides of the equation by log(1.05) to get:
log(105/23.4) / log(1.08) = x
solve for x to get:
x = 19.50628923.
confirm by replacing x in the original equation to get:
h(x) = 23.4 * 1.08 ^ 19.50628923 = 105.
the first year that expenditures reach 105 billion is year 19, which would be 1990 + 19 = 2009.
the actual point is somewhere between 2009 and 2010.
here's a graph of the equation and the important points.
x = 0 = 1990
x = 5 = 1995
x = 19 = 2009
x = 20 = 2010
x = 21 = 2020


<img src = "http://theo.x10hosting.com/2022/012501.jpg" >


let me know if you have any questions.


theo