Final answer:
Using the formula for exponential decay, we can calculate that it will take approximately 8 years for a Nissan Xterra purchased for $25,000 to depreciate to a value of $16,703.58 at an annual rate of 6.5%.
Step-by-step explanation:
To determine after how many years a Nissan Xterra bought for $25,000 will depreciate to $16,703.58 at a rate of 6.5% per year, we can use the formula for exponential decay:
V = P(1 - r)^t
where:
V is the final value we want the car to be, which is $16,703.58,
P is the initial purchase price, which is $25,000,
r is the annual depreciation rate expressed as a decimal, which is 0.065,
and t is the number of years.
To find the number of years (t), we'll rearrange the formula to solve for t:
t = log(V/P) / log(1 - r)
Plugging the numbers in gives us:
t = log(16703.58/25000) / log(1 - 0.065)
Calculating this gives us:
t ≈ 7.91 years
So it will take approximately 8 years for the car to depreciate to $16,703.58.