To determine the value of a car after it has been depreciating for [tex]\( t \)[/tex] years at an annual depreciation rate of 20%, we need to consider how depreciation works.
Depreciation at 20% annually means that each year, the car retains 80% (since [tex]\( 100\% - 20\% = 80\% \)[/tex] or [tex]\( 0.80 \)[/tex]) of its value from the previous year. Therefore, each year, the car's value is multiplied by 0.80.
If we denote the initial value of the car as [tex]\( V_0 \)[/tex] and the value after [tex]\( t \)[/tex] years as [tex]\( V_t \)[/tex], we can express the value after [tex]\( t \)[/tex] years as:
[tex]\[ V_t = V_0 \times (0.80)^t \][/tex]
Here:
- [tex]\( V_0 = 29,873 \)[/tex]
- The annual depreciation factor is [tex]\( 0.80 \)[/tex]
- The number of years is [tex]\( t \)[/tex]
So the expression becomes:
[tex]\[ V_t = 29,873 \times (0.80)^t \][/tex]
Now, we need to compare this derived expression with the given options:
1. [tex]\( 29,873 \times (0.20)^4 \)[/tex]
2. [tex]\( 29,873 \times (20)^t \)[/tex]
3. [tex]\( 29,873 \times (1 - 0.20)^t \)[/tex]
4. [tex]\( 29,873 \times (1 + 0.20)^t \)[/tex]
Option (3) is written as:
[tex]\[ 29,873 \times (1 - 0.20)^t \][/tex]
Since [tex]\( 1 - 0.20 = 0.80 \)[/tex], this option simplifies to:
[tex]\[ 29,873 \times (0.80)^t \][/tex]
This matches our derived expression exactly.
Therefore, the correct expression that can be used to determine the value of the car after [tex]\( t \)[/tex] years is:
[tex]\[ \boxed{3} \][/tex]
[tex]\( 29,873 \times (1 - 0.20)^t \)[/tex]
