Out-of-pocket spending in a country for health care increased between 2003 and 2008 . The function $f(x)=2572 e^{0.0359 x}$ models average annual expenditures per household, in dollars. In this model, $\mathrm{x}$ represents the year, where $\mathrm{x}=0$ corresponds to 2003.
(a) Estimate out-of-pocket household spending on health care in 2008.
(b) Determine the year when spending reached $\$ 2847$ per household.
Answer
Since $x$ represents the year with $x=0$ corresponding to 2003, we add the value of $x$ to 2003 to get the actual year.
Step 1 ：Let's denote the year 2003 as year 0. Therefore, the year 2008 corresponds to year 5. We can find the out-of-pocket household spending on health care in 2008 by substituting $x=5$ into the function $f(x)=2572 e^{0.0359 x}$.
Step 2 ：Calculate $f(5)=2572 e^{0.0359 \times 5}$ to get the estimated out-of-pocket household spending on health care in 2008.
Step 3 ：Next, to find the year when spending reached $2847 per household, we need to solve the equation $f(x) = 2847$ for $x$.
Step 4 ：Solving the equation $2572 e^{0.0359 x} = 2847$ gives us the value of $x$.
Step 5 ：Since $x$ represents the year with $x=0$ corresponding to 2003, we add the value of $x$ to 2003 to get the actual year.
