Sure, let's go through the solution step-by-step.
First, let's make the table of values showing how the car's value changes each year:
| Time (Years) | Value (\[tex]$) |
|--------------|-------------------|
| 0 | 32,000 |
| 1 | 27,200 |
| 2 | 23,120 |
| 3 | 19,652 |
| 4 | 16,704.2 |
| 5 | 14,198.6 |
Explanation of the table:
- At year 0, the value of the car is \$[/tex]32,000.
- At year 1, the value of the car is [tex]\( 32,000 \times 0.85 = 27,200 \)[/tex].
- At year 2, the value of the car is [tex]\( 27,200 \times 0.85 = 23,120 \)[/tex].
- At year 3, the value of the car is [tex]\( 23,120 \times 0.85 = 19,652 \)[/tex].
- At year 4, the value of the car is [tex]\( 19,652 \times 0.85 = 16,704.2 \)[/tex].
- At year 5, the value of the car is [tex]\( 16,704.2 \times 0.85 = 14,198.6 \)[/tex].
Finding the formula:
The general exponential decay formula is given by [tex]\( y = a \cdot b^x \)[/tex], where:
- [tex]\( y \)[/tex] is the value of the car.
- [tex]\( a \)[/tex] is the initial value of the car, which is \[tex]$32,000.
- \( b \) is the rate at which the value is retained each year, which is 0.85 (since it retains 85% of its value each year).
- \( x \) is the time in years.
So, the formula for the car's value in terms of time is:
\[ y = 32,000 \cdot (0.85)^x \]
Finding the value of the car when it is four years old:
To find the value of the car when it is four years old, we substitute \( x = 4 \) into the formula:
\[ y = 32,000 \cdot (0.85)^4 \]
Calculating the value, we get:
\[ y = 32,000 \cdot (0.85)^4 = 32,000 \cdot 0.52200625 \approx 16,704.2 \]
Thus, the value of the car when it is 4 years old is approximately:
\[ \$[/tex]16,704.2 \]
So, the final answer is that at 4 years old, we expect the car to be worth [tex]$16,704.2$[/tex].
The step-by-step detailed solution includes:
1. Creating a table with value calculations over the years.
2. Deriving the general exponential decay formula.
3. Using the formula to find the specific value at 4 years old.
