Question

The total expenditures per household in the year 2008 were approximately $?

Answer 1

Given

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

Solution

Part A

year

`2008-2003=5`

when x=5

`\begin{gathered} f(x)=2577e^(0.0359*5) \\ \\ f(x)=3084 \end{gathered}`

The estimated spending for 2008 is

`\text{ \$3084}`

Part B

when f(x) 2800 find x=?

`2800=2577e^(0.0359x)`

Switch

`\begin{gathered} 2577e^(0.0350x)=2800 \\ divide\text{ both sides by 2577} \\ e^(0.0350x)=(2800)/(2577) \\ \\ e^(0.0350x)=1.08653473 \end{gathered}`

Apply exponent rule

`\begin{gathered} 0.0350x=\ln((2800)/(2577)) \\ \\ divide\text{ both sides by 0.0350} \\ x=(in((2800)/(2577)))/(0.0350) \\ \\ x=2.31179 \end{gathered}`
`x=2.31179\text{years}`