Answer :
Answer:
a) The predicted per capita health expenditures for 2018 is $11943.5.
b) After 2018, it will take 16 years for the 2018 expenditures to double
Step-by-step explanation:
The total national health expenditures per capita, in dollars, is given by the following equation:
[tex]H(t) = 6791e^{0.04343t}[/tex]
In which t is the number of years after 2005.
a. Find the predicted per capita health expenditures for 2018.
2018 is 13 years after 2005, so this is H(13).
[tex]H(t) = 6791e^{0.04343t}[/tex]
[tex]H(13) = 6791e^{0.04343*13}[/tex]
[tex]H(13) = 11943.5[/tex]
The predicted per capita health expenditures for 2018 is $11943.5.
b. According to the model, how long will it take the 2018 expenditures to double?
First we find how many years after 2005, in which t is found when H(t) = 2*11943.5.
So
[tex]H(t) = 6791e^{0.04343t}[/tex]
[tex]2*11943.5 = 6791e^{0.04343t}[/tex]
[tex]e^{0.04343t} = \frac{2*11943.5}{6791}[/tex]
[tex]e^{0.04343t} = 3.5174[/tex]
Applying ln to both sides
[tex]\ln{e^{0.04343t}} = \ln{3.5174}[/tex]
[tex]0.04343t = \ln{3.5174}[/tex]
[tex]t = \frac{\ln{3.5174}}{0.04343}[/tex]
[tex]t = 29[/tex]
It will take 29 years after 2005 = 29-13 = 16 years after 2018.
After 2018, it will take 16 years for the 2018 expenditures to double