The value of a car decreasing at a constant rate of 5% per year can be described by an exponential decay function. Here is a step-by-step explanation detailing how this works:
1. Identify the Initial Value and Rate of Decrease:
- Let's denote the initial value of the car as [tex]\( V_0 \)[/tex]. Suppose the initial value of the car is [tex]$10,000.
- The rate of decrease is 5%, which can be written as a decimal: 0.05.
2. Exponential Decay Formula:
- The formula for exponential decay is:
\[
V(t) = V_0 \times (1 - r)^t
\]
where:
- \( V(t) \) is the value of the car at time \( t \) (in years),
- \( V_0 \) is the initial value of the car,
- \( r \) is the rate of decrease (0.05 for 5%),
- \( t \) is the number of years.
3. Calculate the Value After Each Year:
- After 1 Year:
\[
V(1) = 10000 \times (1 - 0.05)^1 = 10000 \times 0.95 = 9500 \text{ dollars}
\]
- After 2 Years:
\[
V(2) = 10000 \times (1 - 0.05)^2 = 10000 \times 0.95^2 = 10000 \times 0.9025 = 9025 \text{ dollars}
\]
- After 3 Years:
\[
V(3) = 10000 \times (1 - 0.05)^3 = 10000 \times 0.95^3 = 10000 \times 0.857375 = 8573.75 \text{ dollars}
\]
Therefore, the values of the car after 1, 2, and 3 years of ownership are $[/tex]9500, [tex]$9025, and $[/tex]8573.75, respectively.
In conclusion, the type of function that best describes the value of the car based on the number of years it is owned is an exponential decay function.
