Question

Katie received a speeding ticket that incurred a $125 fine. After one month, she will alsobe charged late fees for every month that she is delinquent. The amount she pays ismodeled by an exponential function, and the following table reflects the amount she willhave to pay each month.Month PaidAmount Paid$12512$137.503$151.254$166.38a. Write a function that models the fine Katie will pay in terms of the month it is paid. (5points)b. Using your function, predict how much Katie will pay if she waits until month 12 to payher fine. (5 points)

Answer 1

The amount she pays is modeled by an exponential function given below

$f(x)=a\cdot b^x$

Where a is the initial fine ($125), b is the growth factor, and x is the number of months.

The growth factor is the ratio of any two consecutive terms

$\begin{gathered} b=\frac{137.5}{125}=1.1 \\ b=\frac{151.25}{137.5}=1.1 \\ b=\frac{166.38}{151.25}=1.1 \end{gathered}$

So, the exponential model becomes

$f(x)=125\cdot(1.1)^x$

Part b:

Using your function, predict how much Katie will pay if she waits until month 12 to pay her fine.

Let us substitute x = 12 into the above function

$\begin{gathered} f(x)=125\cdot(1.1)^x \\ f(12)=125\cdot(1.1)^{12} \\ f(12)=125\cdot(3.13843) \\ f(12)=\ 392.30 \end{gathered}$

Therefore, she will have to pay $392.30 if she waits until month 12 to pay her fine.