To determine the value of a car after 5 years, given that it depreciates by 6% per year, we can use the formula for exponential decay:
[tex]\[ V(t) = V_0 \times (1 - r)^t \][/tex]
- [tex]\( V(t) \)[/tex] is the value of the car after [tex]\( t \)[/tex] years.
- [tex]\( V_0 \)[/tex] is the initial value of the car.
- [tex]\( r \)[/tex] is the depreciation rate.
- [tex]\( t \)[/tex] is the number of years.
In this problem, we have:
- [tex]\( V_0 = 31,800 \)[/tex] (initial value of the car)
- [tex]\( r = 0.06 \)[/tex] (depreciation rate)
- [tex]\( t = 5 \)[/tex] (number of years)
Substituting the given values into the formula:
[tex]\[ V(5) = 31800 \times (1 - 0.06)^5 \][/tex]
First, calculate the decrement factor:
[tex]\[ 1 - 0.06 = 0.94 \][/tex]
Next, raise this factor to the power of 5:
[tex]\[ 0.94^5 \approx 0.73391 \][/tex]
Then, multiply this result by the initial value:
[tex]\[ 31800 \times 0.73391 \approx 23338.15 \][/tex]
Therefore, the value of the car after 5 years will be approximately [tex]$23,338.15.
The correct answer is:
C) $[/tex]23,338.15
