To solve this problem, we need to determine after how many years the value of the car will depreciate to [tex]$14,000 or less, starting from an initial value of $[/tex]36,000, with an annual depreciation rate of 20%.
Here’s the step-by-step solution:
1. Understand Initial Values and Depreciation:
- The car's initial value is [tex]$36,000.
- The car depreciates by 20% each year. This means each year the car retains 80% (or 0.80) of its value from the previous year.
2. Set Up the Depreciation Formula:
- Each year, the car's value is multiplied by 0.80 to find its new value.
- Mathematically, if \( V_0 \) is the initial value ($[/tex]36,000), and [tex]\( V_n \)[/tex] is the value after [tex]\( n \)[/tex] years, then:
[tex]\[ V_n = V_0 \times (0.80)^n \][/tex]
3. Determine the Number of Years:
- We need to find the smallest whole number [tex]\( n \)[/tex] such that [tex]\( V_n \leq 14,000 \)[/tex].
4. Calculate Year by Year:
- Year 0: Starting value = [tex]$36,000
- Year 1: \( 36,000 \times 0.80 = 28,800 \)
- Year 2: \( 28,800 \times 0.80 = 23,040 \)
- Year 3: \( 23,040 \times 0.80 = 18,432 \)
- Year 4: \( 18,432 \times 0.80 = 14,745.60 \)
- Year 5: \( 14,745.60 \times 0.80 = 11,796.48 \)
5. Check Against the Required Value:
- After 4 years, the value is approximately $[/tex]14,745.60, which is still more than [tex]$14,000.
- After 5 years, the value is approximately $[/tex]11,796.48, which is less than [tex]$14,000.
Thus, it takes 5 years for the car to be worth $[/tex]14,000 or less. Therefore, the smallest possible whole number answer to the question is 5 years.
