Question

A car that when purchased 5 years ago cost $5,000 has a value now of $900. Find the value of the car of 8 years after its purchase by using the exponential model V(t) = V0etb, in which V(t) is the value of the car at any time t, V0 is the initial cost, t is the time in years, and b is the rate of depreciation. Round your answer to the nearest hundredth.

Answer 1

We will first calculate the value of b using the given infomation:

`V(5)=V_0\cdot e^(5\cdot b)`

`900=5000\cdot e^(5\cdot b)`

`(900)/(5000)=e^(5\cdot b)`

`(9)/(50)=e^(5\cdot b)`

`b=(1)/(5)\log ((9)/(50))`

That must be the rate of depreciation. Now lets find the value of the car 8 years after:

`V(8)=5000\cdot e^{(8)/(5)\cdot\log ((50)/(9))}`

`V(8)\approx321.67`