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Math Help - Euler Car Depreciation Calculus

  1. #1
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    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
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  2. #2
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    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

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