Answer:
10 years until the value of the car is 11000 dollars
Step-by-step explanation:
The value of the car after t years is modeled by the following equation:
[tex]V(t) = V(0)(1-r)^{t}[/tex]
In which V(0) is the initial value and r is the yearly depreciation ratio, as a decimal.
A new car is purchased for 21100 dollars.
This means that [tex]V(0) = 21100[/tex]
The value of the car depreciates at 6% per year.
This means that [tex]r = 0.06[/tex]. So
[tex]V(t) = V(0)(1-r)^{t}[/tex]
[tex]V(t) = 21000(1-0.06)^{t}[/tex]
[tex]V(t) = 21000(0.94)^{t}[/tex]
To the nearest year, how long will it be until the value of the car is 11000 dollars?
This is t when [tex]V(t) = 11000[/tex]
[tex]V(t) = 21000(0.94)^{t}[/tex]
[tex]11000 = 21000(0.94)^{t}[/tex]
[tex](0.94)^{t} = \frac{11000}{21000}[/tex]
[tex](0.94)^{t} = \frac{11}{21}[/tex]
[tex]\log{(0.94)^{t}} = \log{\frac{11}{21}}[/tex]
[tex]t\log{0.94} = \log{\frac{11}{21}}[/tex]
[tex]t = \frac{\log{\frac{11}{21}}}{\log{0.94}}[/tex]
[tex]t = 10.45[/tex]
To the nearest year
10 years until the value of the car is 11000 dollars
