Question

Maureen bought a new car worth $26,525 one year ago. She knows that her car's value will depreciate each year. She uses an online calculator to find that her car is worth $24,403 today.
Write an exponential equation in the form y=a(b)x that can model the value, y, of Maureen's car x years after purchase.
Use whole numbers, decimals, or simplified fractions for the values of a and b.

Answer 1

To write an exponential equation in the form \( y = a(b)^x \) that models the value of Maureen's car over time, we need to identify the values of \( a \) and \( b \).

1. **Identify \( a \)**: The initial value of the car when she bought it (at \( x = 0 \)) is \( a = 26,525 \).

2. **Calculate \( b \)**: To find \( b \), we use the car's value after one year. We know:

- The value of the car after 1 year (\( x = 1 \)) is \( 24,403 \).

We can set up the equation based on the relationship:

\[
y = a(b)^x
\]

Substituting the known values after 1 year:

\[
24403 = 26525(b)^1
\]

To isolate \( b \), we can divide both sides by \( 26525 \):

\[
b = \frac{24403}{26525}
\]

Calculating this fraction:

\[
b \approx 0.920
\]

So the approximate value of \( b \) is \( 0.920 \).

3. **Write the Final Equation**: Now we can write the exponential decay model for the value of the car:

\[
y = 26525(0.920)^x
\]

Thus, the exponential equation that models the value of Maureen's car \( x \) years after purchase is:

\[
y = 26525(0.920)^x
\]