Question

Out of-pocket spending in a country for health care increased between 2003 and 2008. The function fox)-2574models 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 (c) The total expenditures per household in the year 2008 were approximately S (Round to the nearest dollar as needed.)

Answer 1

Answer:

(a) The total expenditures per household in the year 2008 were approximately $3,080.

(b) The spending reached $2875 per household in 2006.

Step-by-step 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.

Answer 2

An exponential model can be found when two data points from the model are known.

The total expenditures per household in the year 2008 were near $3,080.

The spending reached $2875 per household in year 2006

 

(a). Out of-pocket spending in a country for health care given by function,         $f(x) = 2574e^{0.0359x}$

Since $x = 0$ corresponds to 2003, Similarly, x = 5 corresponds to 2008.

Substituting x = 5 into above function to estimate out-of-pocket household spending on health care in 2008

$f(5) = 2574e^{0.0359*5} \\\\f(5)=3080$

f(5) = $3,080

Therefore, the total expenditures per household in the year 2008 were approximately $3,080.

(b). To determine the year when spending reached $2875 per household

 $2875 = 2574e^{0.0359x} \\\\2875/2574=e^{0.0359x} \\\\1.1169=e^{0.0359x}$

 Taking natural log on both side

$x=\frac{ln(1.1169)}{0.0359} =3.07$

Since x = 0 corresponds to 2003, So  x = 3 corresponds to 2006.

Therefore, the spending reached $2875 per household in 2006.

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