Question

The value of a used car can be modeled by the formula V=Vo(1-r)^t where Vo is the car's purchase price, in dollars; r is the car's constant annual rate of decrease in value, expressed as a decimal; and V is the car's dollar value at the end of t years. A used car has a constant annual rate of decrease in value of 0.075. According to the model, what expression would give the number of years after purchase for the car to reach a value that is 50% of its purchase price?

Answer 1

Following the equation

`V(t) = V_0(1-r)^t`

We start with an initial price of

`V(0)=V_0`

and we're looking for a number of years t such that

`V(t)=(V_0)/(2)`

If we substitute V(t) with its equation, recalling that

`r = 0.075 \implies 1-r = 0.925`

we have

`V_0\cdot (0.925)^t=(V_0)/(2) \iff 0.925^t = (1)/(2) \iff t = \log_(0.925)\left((1)/(2)\right)\approx 8.89`

So, you have to wait about 9 years.