Respuesta :
Exponential growth model common ratio is defined as:
[tex]r=(1+k)[/tex]k is the growth rate written in decimals.
Given data:
Pn is the predicted number of speeding tickets during the year 2012+n
P0: 240 (First term)
Growth rate: 15% (0.15)
Then, the common ratio (r) is:
[tex]r=(1+0.15)=1.15[/tex]Recursive formula:
[tex]\begin{gathered} \begin{cases}P_0 \\ P_n=P_{n-1}\cdot r\end{cases} \\ \\ \begin{cases}P_0=240 \\ P_n=P_{n-1}\cdot1.15\end{cases} \\ \end{gathered}[/tex]Explicit formula:
[tex]\begin{gathered} P_n=P_0\cdot r^{n-1} \\ \\ P_n=240\cdot(1.15)^{n-1} \end{gathered}[/tex]If this trend continues, how many speeding tickets are predicted to be issued in 2027:
Identify the value of n corresponding to year 2027:
[tex]\begin{gathered} 2027=2012+n \\ 2027-2012=n \\ \\ n=15 \end{gathered}[/tex]Find the predicted number of speeding tickets during the year 2012+15:
[tex]\begin{gathered} n=15 \\ P_{15}=240\cdot(1.15)^{15-1} \\ P_{15}=240\cdot1.15^{14} \\ P_{15}\approx1698 \end{gathered}[/tex]