Euler Car Depreciation Calculus

Say a car's value drops 20% annually. In ﬁve years estimate

the percentage of the original vehicle value that is retained.

Assume you bought this vehicle at the time t = 0 (assume

the mileage of x = 0 km) for $ 20,000. Assume you drive about

15,000 km annualy (assume uniform driving habits over the year).

Let the input variable x be mileage of the car in km and the

output variable y = f(x) be the value of the car in dollars.

i. Find an expression for the function f(x) in terms of x.

The both equations need to be exponential, but I am having a difficult time figuring it out

Re: Euler Car Depreciation Calculus

well the underlying variable that relates x and f(x) is time.

let $t$ be the years owned

$val(t)=20000(1-0.2)^t=20000(0.8)^t$

$mileage(t)=15000 t$

let $x=mileage(t)$

$x=15000 t$

$t = \dfrac{x}{15000}$

so

$f(x) = val(t(x))=20000(0.8)^{x/15000}$