Question

Sara's new car depreciates 8% each year.

1. Write an exponential function to model the depreciation each month.
2. Write an exponential function to model the depreciation each week.
3. Write an exponential function to model the depreciation each day.
4. What relationship is there between the amount of depreciation and the time interval measured?

Answer 1

If something depreciates, this means its value goes down or is reduced by some factor. The factor here is 8%. So if something is worth Y dollars today, next year it will be worth less:
$$
\large
Y_{\text{nextyear}}=Y-0.08Y=Y(1-.08)=0.92Y
$$
If it depreciates 8% in one year, then each month it depreciates 8/12 % and the value at the end of every month will be:
$$
Y{\text{next month}}=Y\left (1-{.08\over 12}\right)
$$
The month after that it will be worth:
$$
Y{\text{in 2 months}}=Y\left (1-{.08\over 12}\right)^2
$$
After n months:
$$
Y{\text{in n months}}=Y\left (1-{.08\over 12}\right)^n
$$
Similarly for weeks and days. The depreciation per week is 8/52 % and per day is 8/365 % .

Compare the values after a year for each of these to see how increasing the frequency of depreciation affects its value and then you'll have the answer to the last question.