A minivan was purchased for $32,000. If the value of the minivan depreciates by $1,700 per year, find a linear function that models the value V of the car after t years. Use the function and find the value of the car after 5 years.

To model the value of the minivan as it depreciates, we can use the following linear function:

\[
V(t) = P - D \cdot t
\]

Where:
- $V(t)$ is the value of the minivan after $t$ years,
- $P = 32,000$ is the initial purchase price of the minivan,
- $D = 1,700$ is the depreciation rate per year,
- $t$ is the number of years.

### Step 1: Write the linear function
The linear function that models the value $V(t)$ of the car after $t$ years is:

\[
V(t) = 32,000 - 1,700 \cdot t
\]

### Step 2: Calculate the value of the car after 5 years
To find the value of the car after 5 years ($t = 5$), substitute $t = 5$ into the equation:

\[
V(5) = 32,000 - 1,700 \cdot 5
\]

\[
V(5) = 32,000 - 8,500
\]

\[
V(5) = 23,500
\]

Thus, the value of the car after 5 years is **$23,500**.

Would you like further details or have any questions?

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Here are five related questions to expand your understanding:

1. How would the linear function change if the depreciation rate increased to $2,000 per year?
2. After how many years will the value of the minivan be reduced to half its original value?
3. What is the value of the minivan after 10 years of depreciation?
4. If a different car depreciates at a constant percentage rate, what type of function would model its value?
5. What is the slope of the linear function and how does it relate to the problem?

**Tip:** When interpreting linear functions, the slope represents the rate of change, and in depreciation problems, it usually signifies how quickly value decreases over time.

Explore how to model the depreciation of a minivan purchased for $32,000, with a linear function decreasing by $1,700 annually. Learn how to calculate the value of the car after 5 years using algebraic principles. Perfect for students in Grades 8-10.

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