Question

Markus bought a car for $40,000
$
40
,
000
and it decreases in value by 15%
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
𝑥
in your equation. Do not include a comma in your equation. Simplify the expression in the parenthesis.

f(x)=

Answer 1

wrong

Answer 2

To determine the value of the car after each year, we can use the formula for exponential decay. The value of the car decreases by 15% each year, which means it retains 85% of its value at the end of each year.

1. The initial value of the car is \( V_0 = 40,000 \) dollars.
2. The car retains 85% of its value each year, which can be expressed as \( 0.85 \) in the function.
3. Using \( x \) to represent the number of years, the function for the car's value can be written as:

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

Thus, the exponential function that represents the value of the car at the end of every year is:

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

Answer 3

Let's go through the problem step by step again to clarify.

Markus bought a car for $40,000, and it decreases in value by 15% each year.

To express this as an exponential function:

1. Starting value of the car, \( V_0 = 40,000 \).

2. Since it decreases by 15%, it retains 85% of its value each year. Therefore, the factor representing the value retained after each year is:

\[
1 - 0.15 = 0.85
\]

3. The equation for the value of the car after \( x \) years is:

\[
V(x) = V_0 \cdot (1 - \text{decrease rate})^x
\]

Substituting the values:

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

Thus, the correct exponential function representing the value of the car at the end of every year is:

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

If there is a specific aspect of the problem you would like me to address or if there's something you believe is incorrect, please let me know!