A farmer buys a new tractor for ​$146 comma 000 and assumes that it will have a​ trade-in value of ​$79 comma 000 after 10 years. The farmer uses a constant rate of depreciation to determine the annual value of the tractor.
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Part 1
​(A) Find a linear model for the depreciated value V of the tractor t years after it was purchased.
Vequals
  
enter your response here
​(Type your answer in​ slope-intercept form.)

The problem asks for a linear model for the depreciated value $ V $ of the tractor, where $ t $ represents the number of years after the tractor was purchased.

### Step 1: Identify key points.
We are given:
- The tractor is bought for $146,000, meaning at $ t = 0 $, the value $ V(0) = 146,000 $.
- After 10 years, the tractor's trade-in value is $79,000, meaning at $ t = 10 $, $ V(10) = 79,000 $.

These give us two points: $ (0, 146000) $ and $ (10, 79000) $.

### Step 2: Find the slope of the linear model.
The slope $ m $ of the linear equation $ V = mt + b $ can be calculated using the slope formula:
\[
m = \frac{V(10) - V(0)}{10 - 0} = \frac{79000 - 146000}{10 - 0} = \frac{-67000}{10} = -6700
\]

### Step 3: Use the slope-intercept form.
The general equation of a line is $ V = mt + b $, where $ m $ is the slope and $ b $ is the y-intercept. From the given data, we know $ b = 146,000 $, as the initial value of the tractor at $ t = 0 $.

So the equation is:
\[
V = -6700t + 146000
\]

### Final answer:
\[
V(t) = -6700t + 146000
\]

This equation represents the depreciated value of the tractor $ t $ years after it was purchased.

Would you like further details on how this model works, or do you have any specific questions?

### 5 Related Questions:
1. How would the equation change if the tractor's initial value was higher or lower?
2. What happens to the value of the tractor after 15 years?
3. How would you calculate the tractor's value at $ t = 5 $ years using the linear model?
4. Can you use this method for non-linear depreciation models?
5. What if the tractor's trade-in value was lower than expected after 10 years?

### Tip:
Always verify the accuracy of your slope by double-checking the difference in value over time when working with linear depreciation models.

A farmer buys a new tractor for $146,000 and assumes that it will have a trade-in value of $79,000 after 10 years. The farmer uses a constant rate of depreciation to determine the annual value of the tractor. (A) Find a linear model for the depreciated value V of the tractor t years after it was purchased.

Find the linear model for the depreciation of a tractor purchased for $146,000, with a trade-in value of $79,000 after 10 years. Learn how to apply the slope formula and write a linear equation in slope-intercept form. Suitable for students in Grades 9-12.

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