Final answer:
To predict the value of Berenity's car in 1999, we first find the exponential decay rate using the values given for 1995 and 1997. Then, applying this rate, we use the exponential decay formula to calculate the predicted value for 1999.
Step-by-step explanation:
Berenity purchased a new car in 1995 for $24,400 and it depreciated to $13,900 by 1997. The value of the car is depreciating exponentially at a constant rate. To find the predicted value of the car in 1999, we can use the formula for exponential decay, which is:
V = P * e(rt)
Where V is the future value, P is the initial value, e is the base of the natural logarithms, r is the rate of decay, and t is the time in years. Given from the problem we have two points, (0, 24400) where t=0 is the year 1995, and (2, 13900) where t=2 is the year 1997.
First we'll need to find the decay rate 'r' using these two points, solving the equation P * e(rt) = V, we substitute P with 24400, V with 13900, and t with 2 years:
13900 = 24400 * e(2r) → e(2r) = 13900/24400 → 2r = ln(13900/24400) → r = ln(13900/24400) / 2
Once we determine 'r', we will use it to predict the value in 1999, with t equal to 4 since it's 4 years after 1995:
V = 24400 * e(4r)
We then calculate the value to the nearest dollar, and this will give us the predicted value of the car in 1999.