Answer:
4.6% decay
Explanation:
We are asked to model the average price of gas by the following model
[tex]A=Pe^{rt}[/tex]
where t is the number of years after 2006.
Now we are told that when t = 0 ( in 2006), A = 3.92. Putting these values into the above formula gives
[tex]3.92=Pe^{r*0}[/tex][tex]3.92=P[/tex]
With the value of P in hand, our exponential function now becomes
[tex]A=3.92e^{rt}[/tex]
Furthermore, we are also told that when t = 11 ( in 2017), A= $2.36; therefore,
[tex]2.36=3.92e^{11r}[/tex]
Dividing both sides by 3.92 gives
[tex]\frac{2.36}{3.92}=e^{11r}[/tex]
Taking the natural logarithm of both sides gives
[tex]\ln[\frac{2.36}{3.92}]=\ln[e^{11r}][/tex][tex]\ln[\frac{2.36}{3.92}]=11r[/tex]
Evaluating the left-hand side gives
[tex]-0.507...=11r[/tex]
dividing both sides by 11 gives
[tex]r=\frac{-0.507}{11}[/tex][tex]\boxed{r=-0.046.}[/tex]
Hence, our exponential decay function is
[tex]\boxed{A=3.92e^{-0.046t}}[/tex]
From the above calculations, we found that r = -0.046. This means that the percent decay is 0.046 x 100% = 4.6%.
