Question

3. Exponential Functions in Context(1 point)
Markus bought a car for $40,000
and it decreases in value by 15%
every year.

Part A: Write an exponential function that represents the value of the car at the end of every year. Use the variable x
in your equation. Do not include a comma in your equation. Simplify the expression in the parenthesis.

f(x)=


Part B: Fill out the table that represents the value of the car after 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
f(x)

0 $
1 $
3 $
5 $
8 $

Answer 1

### Part A:

To represent the value of the car at the end of every year, we can use the concept of exponential decay. The value of the car decreases by 15% each year, which means it retains 85% of its value (100% - 15% = 85%).

The exponential function can be expressed as:

\[
f(x) = 40000 \cdot (0.85)^x
\]

### Part B:

Now, let's calculate the value of the car for various years (0, 1, 3, 5, and 8) using the function we derived:

1. **For x = 0:**
\[
f(0) = 40000 \cdot (0.85)^0 = 40000 \cdot 1 = 40000.00
\]

2. **For x = 1:**
\[
f(1) = 40000 \cdot (0.85)^1 = 40000 \cdot 0.85 = 34000.00
\]

3. **For x = 3:**
\[
f(3) = 40000 \cdot (0.85)^3 = 40000 \cdot 0.614125 = 24565.00
\]

4. **For x = 5:**
\[
f(5) = 40000 \cdot (0.85)^5 = 40000 \cdot 0.443705 = 17748.20
\]

5. **For x = 8:**
\[
f(8) = 40000 \cdot (0.85)^8 = 40000 \cdot 0.272490 = 10899.60
\]

### Final Table:

| x | f(x) |
|---|--------------|
| 0 | $40000.00 |
| 1 | $34000.00 |
| 3 | $24565.00 |
| 5 | $17748.20 |
| 8 | $10899.60 |

Thus, the values are:
- For \( x = 0 \): $40000.00
- For \( x = 1 \): $34000.00
- For \( x = 3 \): $24565.00
- For \( x = 5 \): $17748.20
- For \( x = 8 \): $10899.60