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8 March, 11:21

Nathaniel purchased a new car in 1997 for $20,300. The value of the car has been depreciating exponentially at a constant rate. If the value of the car was $2,700 in the year 2004, then what would be the predicted value of the car in the year 2007, to the nearest dollar?

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  1. 8 March, 14:54
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    Step-by-step explanation:

    We would apply the formula for exponential decay which is expressed as

    A = P (1 - r) ^t

    Where

    A represents the value of the car after t years.

    t represents the number of years.

    P represents the initial value of the car.

    r represents rate of decay.

    From the information given,

    A = $2700

    P = $20300

    n = 2004 - 1997 = 7 years

    Therefore,

    20300 = 2700 (1 - r) ^7

    20300/2700 = (1 - r) ^7

    7.519 = (1 - r) ^7

    Taking log of both sides, it becomes

    Log 7.519 = 7 log (1 - r)

    0.876 = 7 log (1 - r)

    Log (1 - r) = 0.876/7 = 0.125

    Taking inverse log of both sides, it becomes

    10^log1 - r = 10^0.125

    1 - r = 1.33

    r = 1.33 - 1 = 0.33

    The expression would be

    A = 20300 (1 - 0.33) ^t

    A = 20300 (0.67) ^t

    Therefore, in 2007,

    t = 2007 - 1997 = 10 years

    The value would be

    A = 20300 (0.67) ^10

    A = $370
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