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.