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?

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