SOLUTION: The total expenditures on benefits (in billions of dollars) can be approximated by the function h(x)=23.4(1.08)^x , where x = 5 corresponds to the year 1995. (a)

Question 1190019: The total expenditures on benefits (in billions of dollars) can be approximated by the function h(x)=23.4(1.08)^x , where x = 5 corresponds to the year 1995.
(a) What was the amount for total expenditures in 2011?
(b) What was the first full year in which expenditures exceeded $105 billion?
SOLVE A & B PLEASE
Found 2 solutions by Boreal, Theo:
Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
A. h(16)=23.4(1.08)^21=117.792 billion dollars
B. 105=23.4*1.08^x
divide by the constant
1.08^x=4.487
ln both sides
x ln 1.08=ln 4.487
round at end after dividing both sides by ln 1.08
=19.506 years, so the first full year will be year 20, or 2010
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
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

let me know if you have any questions.

theo

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SOLUTION: The total expenditures on benefits​ (in billions of​ dollars) can be approximated by the function h(x)=23.4(1.08)^x ​, where x​ = 5 corresponds to the year 1995. ​(a)

SOLUTION: The total expenditures on benefits (in billions of dollars) can be approximated by the function h(x)=23.4(1.08)^x , where x = 5 corresponds to the year 1995. (a)