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Let's go step by step with this linear regression problem.

### Problem Overview:
- The truck was bought in 2020.
- In 2023, its value is $30,500.
- The company predicts the truck's value will depreciate linearly until 2033, at which point the salvage value will be $15,500.

The goal is to:
1. Complete the table.
2. Compute the slope of the linear function.
3. Write the equation of the line that models the truck's value over time.

### Step a: Completing the Table
The table needs to show:
- $ t $ (the number of years since 2020)
- $ V $ (the value of the truck)

We know:
- In 2020 ($ t = 0 $), the truck's value was $30,500.
- In 2023 ($ t = 3 $), the value is still $30,500.
- In 2033 ($ t = 13 $), the value will be $15,500.

So, the table would look like this:
| $ t $  | $ V $ (Value in dollars) |
|----------|----------------------------|
| 0        | 30,500                     |
| 3        | 30,500                     |
| 13       | 15,500                     |

### Step b: Compute the Slope of the Linear Function
The slope $ m $ of the line is calculated as the change in value $ \Delta V $ divided by the change in time $ \Delta t $:

\[
m = \frac{\Delta V}{\Delta t} = \frac{15,500 - 30,500}{13 - 0} = \frac{-15,000}{13} = -1153.85 \, \text{dollars per year}
\]

So, the slope is $ -1153.85 $.

### Step c: Write the Equation of the Linear Function
The linear function follows the form:

\[
V(t) = mt + b
\]

Where:
- $ m = -1153.85 $ (the slope)
- $ b = 30,500 $ (the initial value when $ t = 0 $)

Thus, the equation of the line is:

\[
V(t) = -1153.85t + 30,500
\]

---

Would you like more details on any step, or do you have additional questions?

Here are 5 related questions for practice:
1. What would be the value of the truck in 2025 based on this linear model?
2. How long will it take for the truck's value to drop below $10,000?
3. If the truck was purchased for $35,000 instead of $30,500, how would the slope and equation change?
4. How would a different depreciation model, such as exponential, affect the equation for $ V(t) $?
5. Can this linear model accurately predict the value of a truck beyond 2033?

**Tip**: When interpreting the slope, remember that a negative slope indicates depreciation, meaning the value is decreasing over time.

One company buys a new truck in 2020. In 2023, the truck's value was $30,500. The company depreciates the value of the truck linearly over its useful life and determines its salvage value in 2033 is $15,500.

(a) Complete the table where t is the number of years the company has owned the truck, and V is its value.
(b) Compute the slope of the linear function.
(c) Write an equation that models the value of the truck, V, in terms of its age t.

Solve a linear regression problem to model the depreciation of a truck bought in 2020, with a value of $30,500 in 2023 and a predicted salvage value of $15,500 in 2033. Learn to calculate the slope and write the linear equation representing the truck's value over time.

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