Out-of-pocket spending in a country for health care increased between 2002 and 2007. The function f left parenthesis x right parenthesis equals 2579 e Superscript 0.0359 xf(x)=2579e0.0359x models average annual expenditures per​ household, in dollars. In this​ model, x represents the​ year, where xequals=0 corresponds to 2002.
​(a) Estimate​ out-of-pocket household spending on health care in 2007.
​(b) Determine the year when spending reached ​$2862 per household.

Given the exponential model for out-of-pocket healthcare spending per household:

\[
f(x) = 2579e^{0.0359x}
\]

where $ x = 0 $ corresponds to the year 2002.

### Part (a): Estimate out-of-pocket household spending on health care in 2007.

Since $ x = 0 $ corresponds to 2002, $ x = 5 $ will correspond to 2007. We substitute $ x = 5 $ into the model to find the spending in 2007:

\[
f(5) = 2579e^{0.0359 \cdot 5}
\]

Let’s calculate this.

### Part (b): Determine the year when spending reached $2862 per household.

We are given $ f(x) = 2862 $ and need to find the corresponding value of $ x $. Thus, we solve the equation:

\[
2579e^{0.0359x} = 2862
\]

First, divide both sides by 2579:

\[
e^{0.0359x} = \frac{2862}{2579}
\]

Now, take the natural logarithm of both sides to solve for $ x $:

\[
0.0359x = \ln\left(\frac{2862}{2579}\right)
\]

Finally, solve for $ x $:

\[
x = \frac{\ln\left(\frac{2862}{2579}\right)}{0.0359}
\]

I’ll calculate both parts now.### Part (a): The estimated out-of-pocket household spending on health care in 2007 is approximately **$3086.08**.

### Part (b): The spending reached $2862 per household in the year **2004.9**, which is roughly towards the end of **2004**.

Do you need further clarification or more details on these steps?

Here are 5 related questions you might consider:
1. How does the exponential growth model compare to a linear growth model in this context?
2. What would the spending be in 2010 using this model?
3. How would a change in the rate of growth (0.0359) affect future predictions?
4. Can you derive the formula to predict spending for any given year in terms of $ x $?
5. What are the limitations of using an exponential model for long-term predictions?

**Tip:** When solving exponential equations, using logarithms is the key to isolating the variable in the exponent.

Out-of-pocket spending in a country for health care increased between 2002 and 2007. The function f(x) = 2579e^0.0359x models average annual expenditures per household, in dollars. In this model, x represents the year, where x = 0 corresponds to 2002. (a) Estimate out-of-pocket household spending on health care in 2007. (b) Determine the year when spending reached $2862 per household.

Learn how to estimate out-of-pocket healthcare spending using an exponential growth model. This problem involves solving exponential equations and logarithmic calculations. Perfect for students in Grades 10-12 studying algebra and growth models.

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