The function representing the value of the car is given to be:
[tex]C(t)=23746\cdot e^{(rt)}[/tex]
After 2 years, the value is $20,233. This means that:
[tex]\begin{gathered} when \\ t=2,C(t)=20233 \end{gathered}[/tex]
Therefore, the value of r can be calculated to be:
[tex]\begin{gathered} 20233=23746\cdot e^{(2r)} \\ \mathrm{Divide\:both\:sides\:by\:}23746 \\ \frac{23746e^{2r}}{23746}=\frac{20233}{23746} \\ e^{2r}=\frac{20233}{23746} \\ \text{Applying exponent rules} \\ 2r=\ln \left(\frac{20233}{23746}\right) \\ r=\frac{\ln\left(\frac{20233}{23746}\right)}{2} \\ \therefore \\ r=-0.08 \end{gathered}[/tex]
Therefore, the function becomes:
[tex]C(t)=23746\cdot e^{-0.08t}[/tex]
Therefore, after 7 years, the car's worth would be:
[tex]\begin{gathered} At\text{ }t=7 \\ C(t)=23746\cdot e^{-0.08(7)} \\ C(t)\approx13,563.93 \end{gathered}[/tex]
Hence, the correct option is OPTION B.
