Discover new information and insights with the help of IDNLearn.com. Our platform provides detailed and accurate responses from experts, helping you navigate any topic with confidence.
Sagot :
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.
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.
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Your search for answers ends at IDNLearn.com. Thank you for visiting, and we hope to assist you again soon.