Question

During the year ? , spending reached $2800 per household

Answer 1

The function that models average annual expenditures per household, in dollars.

`f(x)=2577e^(0.0359x)`

Where x represents the year, where x = 0 corresponds to 2003.

`\begin{gathered} x=0..........2003 \\ \\ x=1...........2004 \\ \\ x=2.............2005 \\ \\ x=3.............2006 \\ \\ x=4..............2007 \\ \\ x=5..............2008 \end{gathered}`

Thus;

`\begin{gathered} f(5)=2577e^(0.0359(5)) \\ \\ f(5)=3083.69 \\ \\ f(5)\approx3084 \end{gathered}`

ANSWER: $3084

(b)

`\begin{gathered} f(x)=2800 \\ \\ 2800=2577e^(0.0359x) \\ \\ (2800)/(2577)=(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 \\ \\ (0.0830)/(0.0359)=(0.0359x)/(0.0359) \\ \\ x=2.31 \\ \\ x\approx2 \end{gathered}`

The annual spending reached $2800 per household during the year 2005.