Question

In 2012 your car was worth $10,000. In 2014 your car was worth $8,850. Suppose the value of your car decreased at a constant rate of change. Define a function f to determine the value of your car (in dollars) in terms of the number of years t since 2012.

Answer 1

The function to determine the value of your car (in dollars) in terms of the number of years t since 2012 is:

`f(t) = 10000(0.9407)^t`

Explanation:

Value of the car:

Constant rate of change, so the value of the car in t years after 2012 is given by:

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

In which f(0) is the initial value and r is the decay rate, as a decimal.

In 2012 your car was worth $10,000.

This means that
`f(0) = 10000`, thus:

`f(t) = 10000(1-r)^t`

2014 your car was worth $8,850.

2014 - 2012 = 2, so:

`f(2) = 8850`

We use this to find 1 - r.

`f(t) = 10000(1-r)^t`

`8850 = 10000(1-r)^2`

`(1-r)^2 = (8850)/(10000)`

`(1-r)^2 = 0.885`

`√((1-r)^2) = √(0.885)`

`1 - r = 0.9407`

Thus

`f(t) = 10000(1-r)^t`

`f(t) = 10000(0.9407)^t`