Answer:
(a) The total expenditures per household in the year 2008 were approximately $3,080.
(b) The spending reached $2875 per household in 2006.
Explanation:
Note: This question is not complete, and the part c is not another question but the same question as part a. The complete question is therefore provided before answering the question as follows:
Out of-pocket spending in a country for health care increased between 2003 and 2008. The function f(x) = 2574e^(0.0359x) models average annual expenditures per household, in dollars. In this model, x represents the year, where x = 0 corresponds to 2003.
(a) Estimate out-of-pocket household spending on health care in 2008.
(b) Determine the year when spending reached $2875 per household
The explanation to the answer is now given as follows:
(a) Estimate out-of-pocket household spending on health care in 2008.
Given;
f(x) = 2574e^(0.0359x) ........................ (1)
Since x = 0 corresponds to 2003; by counting from 2003 to 2008, it implies that x = 5 corresponds to 2008.
Substituting x = 5 into equation (1) to estimate out-of-pocket household spending on health care in 2008, we have:
f(6) = 2574e^(0.0359 * 5)
f(6) = 2574 * 1.19661890406748
f(6) = 3,080.09705906969
Rounding to the nearest dollar as needed, we have:
f(6) = $3,080
Therefore, the total expenditures per household in the year 2008 were approximately $3,080.
(b) Determine the year when spending reached $2875 per household
This can be determined using equation (1) in part a by equating f(x) to $2875 solve for x as follows:
2875 = 2574e^(0.0359x)
2875 /2574 = e^(0.0359x)
1.11693861693862 = e^(0.0359x)
Taking the natural log of both sides, we have:
ln(1.11693861693862) = ln(e^(0.0359x))
0.110591565075382 = 0.0359x
x = 0.110591565075382 / 0.0359
x = 3.0805449881722
Approximating to a whole number, we have:
x = 3
Since x = 0 corresponds to 2003; by counting from 2003, x = 3 corresponds to 2006.
Therefore, the spending reached $2875 per household in 2006.