3. Exponential Functions in Context(1 point)

Question

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