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Let's solve the problem step by step.

### Part A: Writing the Exponential Function

The value of a new pickup truck depreciates by 9% per year. The initial value of the truck in 2017 is $41,000. The exponential decay model for the value of the truck after $ x $ years can be written as:

\[
y = A \cdot (1 - r)^x
\]

Where:
- $ y $ is the value of the truck after $ x $ years,
- $ A $ is the initial value of the truck ($41,000),
- $ r $ is the depreciation rate (9% or 0.09).

Substituting the values:

\[
y = 41000 \cdot (1 - 0.09)^x
\]

Simplifying further:

\[
y = 41000 \cdot (0.91)^x
\]

### Part B: Finding When the Truck's Value Drops to $20,000

To find the number of years $ x $ it takes for the truck to be worth $20,000, we set $ y = 20000 $ and solve for $ x $:

\[
20000 = 41000 \cdot (0.91)^x
\]

First, divide both sides by 41000:

\[
\frac{20000}{41000} = (0.91)^x
\]

Simplifying the fraction:

\[
\frac{20000}{41000} \approx 0.4878
\]

So,

\[
0.4878 = (0.91)^x
\]

To solve for $ x $, take the logarithm of both sides:

\[
\log(0.4878) = x \cdot \log(0.91)
\]

Now, solve for $ x $:

\[
x = \frac{\log(0.4878)}{\log(0.91)}
\]

Using a calculator to find the logarithms:

\[
x \approx \frac{-0.3118}{-0.0414} \approx 7.5
\]

So, it will take approximately 7.5 years for the truck to depreciate to $20,000.

### Final Answers:
- **Part A:** The function is $ y = 41000 \cdot (0.91)^x $.
- **Part B:** The truck will be worth $20,000 after approximately **7.5 years**.

Would you like more details or have any questions? Here are some related questions for further practice:

1. What would be the value of the truck after 10 years?
2. How long will it take for the truck to depreciate to half of its original value?
3. If the depreciation rate was 12% instead of 9%, how would the function change?
4. Calculate the value of the truck after 5 years.
5. What is the percentage decrease in value after 3 years?

**Tip:** Always remember to apply logarithms carefully when solving exponential equations to find the variable in the exponent!

The moment you drive a car off a dealer's lot, it begins to depreciate, or lose its value. In 2017, the average price of a new pickup truck was $41,000. You can use an exponential function to model the value of a pickup truck over time.

Part A: If the value of a pickup truck depreciates by 9% per year, write a function that represents the value of a new 2017 pickup truck y after x years.

Part B: After how many years will a new 2017 pickup truck be worth only $20,000? Round to the nearest tenth if necessary.

Learn how to model the depreciation of a pickup truck using exponential functions. Solve for the number of years it takes for the truck's value to drop to a certain amount, with detailed explanations of exponential decay and logarithmic calculations. Suitable for students in Grades 9-11.

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