Samantha purchased a new car in 2010 for $16,200. The value of the car has been depreciating exponentially at a constant rate. If the value of the car was $9,300 in the year 2012, then what would be the predicted value of the car in the year 2020, to the nearest dollar?

9514 1404 393

  $1,010

Step-by-step explanation:

We can write the exponential decay function in the form ...

  y = a(b^(t/T))

where 'a' is the initial value, and 'b' is the "growth" factor in time period T.

Here, the initial value is 16200, for t=0 corresponding to the year 2010.

The "growth" in value is by the factor 9300/16200 in the 2-year period between 2010 and 2012. This means we can write the exponential function as ...

  y = 16200((93/162)^(t/2))

In the year 2020, the value of t is 2020 -2010 = 10, so the car value will be ...

  y = 16200(93/162)^(10/2) = 16200(93/162)^5 ≈ 1010

The predicted value in 2020 is $1,010.

If you like, you can "simplify" the base of the exponential factor to ...

  (93/162)^(1/2) ≈ 0.75768

Then the value function becomes ...

  y = 16,200(0.75768^t)

the answer to this question is in the pic