Question

ill out the table that represents the value of the car after x
𝑥
years. Do not include a comma in your response. Write your answer to the nearest cent, write ".00" if there are no cents.

x
𝑥
f(x)
𝑓
(
𝑥
)

0 $
1 $
3 $
5 $
8 $

Answer 1

To complete the table representing the value of a car after x years, typically, we would use a formula for depreciation. A common model for car depreciation is that a car loses a fixed percentage of its value each year. For this example, let's assume the car starts with an initial value of $20,000 and depreciates at a rate of 15% per year.

Using the formula for depreciation:

\[
f(x) = P(1 - r)^x
\]

where:
- \( P \) is the initial value of the car (here, \( 20000 \)),
- \( r \) is the rate of depreciation (15% or 0.15),
- \( x \) is the number of years.

Let's calculate \( f(x) \) for different values of \( x \):

1. **For \( x = 0 \)**:
\[
f(0) = 20000 \times (1 - 0.15)^0 = 20000 \times 1 = 20000.00
\]

2. **For \( x = 1 \)**:
\[
f(1) = 20000 \times (1 - 0.15)^1 = 20000 \times 0.85 = 17000.00
\]

3. **For \( x = 3 \)**:
\[
f(3) = 20000 \times (1 - 0.15)^3 = 20000 \times 0.85^3 \approx 20000 \times 0.614125 = 12282.50
\]

4. **For \( x = 5 \)**:
\[
f(5) = 20000 \times (1 - 0.15)^5 = 20000 \times 0.85^5 \approx 20000 \times 0.443705 = 8874.10
\]

5. **For \( x = 8 \)**:
\[
f(8) = 20000 \times (1 - 0.15)^8 = 20000 \times 0.85^8 \approx 20000 \times 0.272490 = 5449.80
\]

Now we can fill out the table:

\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
0 & 20000.00 \\
1 & 17000.00 \\
3 & 12282.50 \\
5 & 8874.10 \\
8 & 5449.80 \\
\hline
\end{array}
\]

The completed values are:

- For \( x = 0 \): **$20000.00**
- For \( x = 1 \): **$17000.00**
- For \( x = 3 \): **$12282.50**
- For \( x = 5 \): **$8874.10**
- For \( x = 8 \): **$5449.80**