A farm purchased a combine harvester valued at 
\[\$330{,}000\]. The value of the machine depreciates by the same amount each year so that after 
\[10\] years the value will be 
\[\$80{,}000\]. Which of the following equations gives the value, 
\[v\], of the harvester, in dollars, 
\[t\] years after it was purchased for 
\[0 \leq t \leq 10\] ?

To solve this problem, we need to determine the equation that represents the value $ v $ of the harvester $ t $ years after it was purchased. The harvester's value depreciates linearly over time, meaning the value decreases by a constant amount each year.

### Step 1: Define the Linear Equation

The value of the harvester can be modeled by a linear equation of the form:
\[
v(t) = mt + b
\]
where:
- $ v(t) $ is the value of the harvester after $ t $ years,
- $ m $ is the slope (rate of depreciation per year),
- $ b $ is the initial value of the harvester.

### Step 2: Determine the Slope $ m $

The slope $ m $ represents the rate of depreciation. We know:
- The initial value (when $ t = 0 $) is $ v(0) = 330{,}000 $ dollars.
- The value after 10 years (when $ t = 10 $) is $ v(10) = 80{,}000 $ dollars.

The slope $ m $ is calculated as:
\[
m = \frac{v(10) - v(0)}{10 - 0} = \frac{80{,}000 - 330{,}000}{10} = \frac{-250{,}000}{10} = -25{,}000
\]
So, the harvester depreciates by $ 25{,}000 $ dollars per year.

### Step 3: Determine the y-intercept $ b $

The y-intercept $ b $ is the value of the harvester when $ t = 0 $, which is the initial value:
\[
b = 330{,}000
\]

### Step 4: Write the Final Equation

Now that we have $ m = -25{,}000 $ and $ b = 330{,}000 $, the equation for the value $ v $ of the harvester after $ t $ years is:
\[
v(t) = -25{,}000t + 330{,}000
\]

This equation is valid for $ 0 \leq t \leq 10 $.

### Answer

The correct equation is:
\[
v = -25{,}000t + 330{,}000
\]

Would you like further details or have any questions? Here are some related questions you might find useful:

1. How do you calculate the depreciation per year using different methods?
2. What would be the value of the harvester after 5 years?
3. Can the depreciation rate be different each year, and how would that affect the equation?
4. What if the initial value of the harvester was different? How would that change the equation?
5. How can you graph this linear equation to visualize the depreciation over time?
6. What are some other real-life examples where linear depreciation might apply?
7. How would the equation change if the harvester had a residual value at the end of 10 years?
8. How is this problem related to concepts in financial mathematics?

**Tip:** Always check the units and ensure they match the context of the problem.

Understand how to calculate depreciation of a combine harvester valued at $330,000 over a period of 10 years. Learn the linear equation model v(t) = -25,000t + 330,000, where v(t) is the harvester's value after t years.

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