Question

g The total national health expenditures per capita (in dollars) for 2006 and projected to 2021 can be modeled with the equation H(t) = 6791e0.04343t Where to is the number of years past 2005. a. Find the predicted per capita health expenditures for 2018. b. According to the model, how long will it take the 2018 expenditures to double?

Answer 1

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

Explanation:

The total national health expenditures per capita, in dollars, is given by the following equation:

`H(t) = 6791e^(0.04343t)`

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).

`H(t) = 6791e^(0.04343t)`

`H(13) = 6791e^(0.04343*13)`

`H(13) = 11943.5`

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

`H(t) = 6791e^(0.04343t)`

`2*11943.5 = 6791e^(0.04343t)`

`e^(0.04343t) = (2*11943.5)/(6791)`

`e^(0.04343t) = 3.5174`

Applying ln to both sides

`\ln{e^(0.04343t)} = ln(3.5174)`

`0.04343t = ln(3.5174)`

`t = (ln(3.5174))/(0.04343)`

`t = 29`

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