Question

A cars value is declining exponentially. The car is currently 3 years old and has a value of 18,000. The car sold for 26,000 brand new. Find the rate at which the value of the car is decreasing.

So far my equation is 18,000=26,000 (3-.04)^the
But I'm not sure if that is right

Answer 1

you know that an exponential function for the value v after t years looks like

v = a e^(-kt)
at t=0, v=26000, so

v = 26000 e^(-kt)
v(3) = 18000, so

26000 e^(-3k) = 18000
e^(-3k) = 9/13
-3k = ln(9/13)
k = 0.1226

v(t) = 26000 e^(-0.1226t)

That's all well and good, but what's the percentage rate?

You know there's a constant ratio from year to year, so

v(t+1)/v(t) = e^-.1226 = 0.88

so, the value declines by 12% each year.

or, knowing the yearly sequence of values forms a geometric progression,

r^3 = 9/13
r = 0.88
as above

Answer 2

v = Vi c^t
.692 = c^3
c = .692^(1/3)
c = .884

every year the car loses (1-.884) = .116 or 11.6% of its value

value = initial value * .884^t