A car is purchased for $24,000 . Each year it loses 30% of its value. After how many years will t

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  1. math_celebrity

    math_celebrity Administrator Staff Member

    A car is purchased for $24,000 . Each year it loses 30% of its value. After how many years will the car be worth $7300 or less? (Use the calculator provided if necessary.) Write the smallest possible whole number answer.

    Set up the depreciation equation D(t) where t is the number of years in the life of the car:
    D(t) = 24,000/(1.3)^t

    The problem asks for D(t)<=7300
    24,000/(1.3)^t = 7300

    Cross multiply:
    7300(1.3)^t = 24,000

    Divide each side by 7300
    1.3^t = 24000/7300
    1.3^t = 3.2877

    Take the natural log of both sides:
    LN(1.3^t) = LN(3.2877)

    Using the natural log identities, we have:
    t * LN(1.3) = 1.1902
    t * 0.2624 = 1.1902

    Divide each side by 0.2624
    t = 4.5356
    Rounding this up, we have t = 5
     

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