Question

A new car is purchased for 21100 dollars. The value of the car depreciates at

6% per year. To the nearest year, how long will it be until the value of the car

is 11000 dollars?

Answer 1

11

Explanation:

Answer 2

10 years until the value of the car is 11000 dollars

Explanation:

The value of the car after t years is modeled by the following equation:

`V(t) = V(0)(1-r)^(t)`

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
`V(0) = 21100`

The value of the car depreciates at 6% per year.

This means that
`r = 0.06`. So

`V(t) = V(0)(1-r)^(t)`

`V(t) = 21000(1-0.06)^(t)`

`V(t) = 21000(0.94)^(t)`

To the nearest year, how long will it be until the value of the car is 11000 dollars?

This is t when
`V(t) = 11000`

`V(t) = 21000(0.94)^(t)`

`11000 = 21000(0.94)^(t)`

`(0.94)^(t) = (11000)/(21000)`

`(0.94)^(t) = (11)/(21)`

`\log{(0.94)^(t)} = \log{(11)/(21)}`

`t\log{0.94} = \log{(11)/(21)}`

`t = \frac{\log{(11)/(21)}}{\log{0.94}}`

`t = 10.45`

To the nearest year

10 years until the value of the car is 11000 dollars