Question

Joyce bought a $28,625 new car which depreciates at 9.7% per year. Find the car's valueafter 10 years.

Answer 1

We know the initial value of the car at time t = 0 and we know that it depreciates 9.7% per year.

If y is the value of the car and t is the time in years, we can write that:

$\begin{gathered} y(t+1)=y(t)-0.097y(t)_{} \\ y(t+1)=(1-0.097)\cdot y(t) \\ y(t+1)=0.903\cdot y(t) \\ \frac{y(t+1)}{y(t)}=0.903 \end{gathered}$

This correspond to an exponential decay model, that can be expressed as:

$y=a\cdot b^t$

The parameter a can be calculated as:

$\begin{gathered} y(0)=a\cdot b^0 \\ y(0)=a \\ 28625=a \end{gathered}$

We can use the previous relationship to find the value of b:

$\begin{gathered} \frac{y(t+1)}{y(t)}=0.903 \\ \frac{a\cdot b^{t+1}}{a\cdot b^t}=0.903 \\ b^{t+1-t}=0.903 \\ b=0.903 \end{gathered}$

Then, we can express the value of the car after t years as:

$y(t)=28625\cdot0.903^t$

Then, after t = 10 years, the value is:

$\begin{gathered} y(10)=28625\cdot0.903^{10} \\ y(10)\approx28625\cdot0.360477 \\ y(10)\approx10318.65 \end{gathered}$

Answer: after 10 years, the value of the car is $10,318.65.