Question

A new car is purchased for dollar sign, 35, comma, 000$35,000 and over time its value depreciates by one half every 3 years. How long, to the nearest tenth of a year, would it take for the value of the car to be dollar sign, 200$200

Answer 1

22.4 years

Explanation:

You want the number of years it takes a $35,000 car to depreciate in value to $200 if the value is cut in half every 3 years.

Model

The exponential decay in value can be modeled by the function ...

v(t) = (initial value)(decay factor)^(t/(decay period))

When the initial value is 35000, the decay factor is 1/2, and the period associated with that is 3 years, this becomes ...

v(t) = 35000(1/2)^(t/3)

Time

We want the value of t when v(t) = 200.

200 = 35000(1/2)^(t/3)

200/35000 = (1/2)^(t/3)

Taking logarithms, we have ...

ln(200/35000) = (t/3)ln(1/2)

Then the value of t is ...

t = 3(ln(200/35000)/ln(1/2)) ≈ 22.4 . . . . years

It would take about 22.4 years for the value to depreciate to $200.

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