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
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
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
To find the rate at which the value of the car is decreasing, you can use the exponential decay formula. However, the equation you provided is not correct. Let's break down the problem and solve it step by step.
Given:
- The car is currently 3 years old and has a value of $18,000.
- The car sold for $26,000 brand new.
First, let's define the equation for exponential decay. The formula for exponential decay is:
V = A * e^(-kt)
Where:
- V is the current value of the car.
- A is the initial value of the car.
- e is the base of the natural logarithm (approximately 2.71828).
- k is the decay constant.
- t is the time in years.
To find the decay constant (k), we substitute the known values into the equation. At t = 0, the car's value was $26,000, so we have:
26,000 = A * e^(-k * 0)
Since e^0 equals 1, we can simplify the equation to:
26,000 = A
Now we can substitute the current value of the car (V = $18,000) and solve for the decay constant (k) and the initial value of the car (A):
18,000 = 26,000 * e^(-k * 3)
Divide both sides of the equation by 26,000:
18,000 / 26,000 = e^(-3k)
0.6923 = e^(-3k)
Now, we need to take the natural logarithm (ln) of both sides of the equation to isolate the decay constant (k):
ln(0.6923) = ln(e^(-3k))
ln(0.6923) = -3k
To find k, divide both sides of the equation by -3:
k ≈ -0.231
Therefore, the decay constant (k) is approximately -0.231.
The rate at which the value of the car is decreasing is equal to the absolute value of the decay constant (k), which in this case would be approximately 0.231 or 23.1%.
Please note that the negative sign in the decay constant indicates the decrease in value over time. However, when calculating the rate of decrease, we take the absolute value to get a positive percentage.