Question

Starting in the year 2012, the number of speeding tickets issued each year in Middletown is predicted to grow according to an exponential growth model. During the year 2012, Middletown issued 240 speeding tickets (P0=240). Every year thereafter, the number of speeding tickets issued is predicted to grow by 15%.If Pn denotes the predicted number of speeding tickets during the year 2012+n, then Write the recursive formula for Pn.Pn_______x Pn-1Write the explicit formula for Pn.Pn___________If this trend continues, how many speeding tickets are predicted to be issued in 2027?____________tickets

Answer 1

Exponential growth model common ratio is defined as:

$r=(1+k)$

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:

$r=(1+0.15)=1.15$

Recursive formula:

$\begin{gathered} \begin{cases}P_0 \\ P_n=P_{n-1}\cdot r\end{cases} \\ \\ \begin{cases}P_0=240 \\ P_n=P_{n-1}\cdot1.15\end{cases} \\ \end{gathered}$

Explicit formula:

$\begin{gathered} P_n=P_0\cdot r^{n-1} \\ \\ P_n=240\cdot(1.15)^{n-1} \end{gathered}$

If this trend continues, how many speeding tickets are predicted to be issued in 2027:

Identify the value of n corresponding to year 2027:

$\begin{gathered} 2027=2012+n \\ 2027-2012=n \\ \\ n=15 \end{gathered}$

Find the predicted number of speeding tickets during the year 2012+15:

$\begin{gathered} n=15 \\ P_{15}=240\cdot(1.15)^{15-1} \\ P_{15}=240\cdot1.15^{14} \\ P_{15}\approx1698 \end{gathered}$

Then, in year 2027 the number of speeding tickets is: 1698