Solution:
The function that models average annual expenditures per household, in dollars.
[tex]f(x)=2577e^{0.0359x}[/tex]
Where x represents the year, where x = 0 corresponds to 2003.
[tex]\begin{gathered} x=0..........2003 \\ \\ x=1...........2004 \\ \\ x=2.............2005 \\ \\ x=3.............2006 \\ \\ x=4..............2007 \\ \\ x=5..............2008 \end{gathered}[/tex]
Thus;
[tex]\begin{gathered} f(5)=2577e^{0.0359(5)} \\ \\ f(5)=3083.69 \\ \\ f(5)\approx3084 \end{gathered}[/tex]
ANSWER: $3084
(b)
[tex]\begin{gathered} f(x)=2800 \\ \\ 2800=2577e^{0.0359x} \\ \\ \frac{2800}{2577}=\frac{2577e^{0.0359x}}{2577} \\ \\ 1.0865=e^{0.0359x} \\ \\ \text{ Take the logarithm of both sides;} \\ \\ \ln(1.0865)=\ln(e^{0.0359x}) \\ \\ 0.0830=0.0359x \\ \\ \frac{0.0830}{0.0359}=\frac{0.0359x}{0.0359} \\ \\ x=2.31 \\ \\ x\approx2 \end{gathered}[/tex]
The annual spending reached $2800 per household during the year 2005.
