91Ó°ÊÓ

As the starting year is 2006, we have to calculate the proportion of new cars at the initial moment, i.e. for $t=0$

$$\begin{gathered} IfP\left(t\right)=\frac{0.8}{1+1.67e^{-0.16t}} \\ thenP\left(0\right)=\frac{0.8}{1+1.67e^{-0.16·0}} \\ P\left(0\right)=0.3=30\% \end{gathered}$$

In 2006, 30% of new cars had GPS. (Or we can say the proportion of new cars that had GPS in 2006 is 0.3)

The maximum proportion of new cars that have GPS is actually the carrying capacity in the formula

$P\left(t\right)=\frac{0.8}{1+1.67e^{-0.16t}}$

The logistic model is given by function

$P\left(t\right)=\frac{c}{1+a{e}^{-bt}},$

where the constant c represents carrying capacity.

Looking at our model we conclude that the carrying capacity is $c=0.8$

Conclusion:The maximum proportion of new cars that have a GPS is 0.8. (A maximum of 80% of new cars can have GPS in 2006)

How do we draw

$P\left(t\right)=\frac{0.8}{1+1.67e^{-0.16t}}$

graphically using a calculator? (We will explain on the example of the calculator Ti-84 Plus C)

1.) Press Y =

2.) Write: $Y_1=\frac{0.8}{1+1.67e^{-0.16x}}$

3.) Press GRAPH

We adjust $X_{min}and{X}_{max}$(press the WINDOW key )

(We can also adjust the values on the y-axis)

$$\begin{gathered} X_{min}=-2 \\ {X}_{max}=30 \\ {Y}_{min}=-2 \\ {Y}_{max}=1(duetocarryingcapacity,c=0.8) \end{gathered}$$

Graph of $P=P\left(t\right)$is:

We know that the equation is given by:

$P\left(t\right)=\frac{0.8}{1+1.67e^{-0.16t}}$

where $t$represent time in years.

We need to find the time $t$in years for which will be $P\left(t\right)=75\%=0.75$

If $P\left(t\right)=0.75$then we need to calculate $t$:

$$\begin{gathered} P\left(t\right)=\frac{0.8}{1+1.67e^{-0.16t}} \\ 0.75=\frac{0.8}{1+1.67e^{-0.16t}} \\ 1+1.67e^{-0.16t}=\frac{0.8}{0.75} \\ 1.67e^{-0.16t}=\frac{0.8}{0.75}-1 \\ 1.67e^{-0.16t}=\frac{0.05}{0.75} \\ 1.67e^{-0.16t}=\frac{1}{15} \\ {e}^{-0.16t}=\frac{20}{501} \\ IfM=Nthen{\log }_{a}M={\log }_{a}N \\ ln{e}^{-0.16t}=\ln \frac{20}{501} \\ Weknowthat{\log }_a{a}^r=r \\ -0.16t=\ln \frac{20}{501} \\ t=\frac{\ln {\displaystyle \frac{20}{501}}}{-0.16} \\ t=20.13years \end{gathered}$$

20.13 years after 2006: During 2026 75% of new cars will have a GPS.