Question

. Sydney bought a new car for $32,000.00
.

If the car depreciates in a value by 8%
each year, the situation can be modeled by the equationf(x)=32,000(0.92)x

Complete the table to find the value of the car to the nearest cent after five years.

x f(x)
0 $32,000.00
1 $29,440.00
2 $27,048.80
5 ?
(1 point)
The value of the car after three years is $
(round your answer to the nearest cent)
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Answer 1

To find the value of the car after five years using the formula \( f(x) = 32,000(0.92)^x \), we need to calculate \( f(5) \).

The formula for the value of the car after \( x \) years is:
\[ f(x) = 32,000(0.92)^x \]

### Step 1: Calculate \( f(5) \)
Substituting \( x = 5 \) into the equation:
\[
f(5) = 32,000(0.92)^5
\]

Calculating \( (0.92)^5 \):
\[
(0.92)^5 \approx 0.659081
\]

Now, substituting this value back into the equation for \( f(5) \):
\[
f(5) = 32,000 \times 0.659081 \approx 21,091.97
\]

### Step 2: Complete the table
Now we can complete the table for \( x = 5 \):

| x | f(x) |
|---|--------------|
| 0 | $32,000.00 |
| 1 | $29,440.00 |
| 2 | $27,048.80 |
| 5 | $21,091.97 |

### Step 3: Find the value of the car after three years
Now, let’s also calculate \( f(3) \):
\[
f(3) = 32,000(0.92)^3
\]

Calculating \( (0.92)^3 \):
\[
(0.92)^3 \approx 0.778688
\]

Now substituting this value back into the equation for \( f(3) \):
\[
f(3) = 32,000 \times 0.778688 \approx 24,971.00
\]

### Final answer
The value of the car after three years is \( \approx \$24,971.00 \) (rounded to the nearest cent).

### Summary
- Value after five years: **$21,091.97**
- Value after three years: **$24,971.00**