Suppose that a company has just purchased a new computer for ​$2400. The company chooses to depreciate using the​ straight-line method for 3 years.
​(a) Write a linear function that expresses the book value V of the computer as a function of its age x.
​(b) What is the domain of the function found in part ​(a)​?
​(c) Graph the linear function.
​(d) What is the book value of the computer after 2 ​years?
​(e) When will the computer have a book value of $ 1600$

Let’s solve this step-by-step.

### (a) Writing the linear function for depreciation:

The **straight-line depreciation method** reduces the value of an asset by a fixed amount each year. For this problem:

- The initial value of the computer is $ \$2400 $.
- The depreciation period is 3 years.
- At the end of 3 years, the computer’s value will be $ \$0 $.

The depreciation per year is:

\[
\text{Depreciation rate} = \frac{\$2400 - 0}{3} = \$800 \, \text{per year}
\]

The book value $ V $ of the computer decreases by \$800 each year, so the linear function representing the book value $ V(x) $ as a function of its age $ x $ (in years) is:

\[
V(x) = 2400 - 800x
\]

Where:
- $ V(x) $ is the book value after $ x $ years.
- $ x $ is the age of the computer in years.

### (b) Domain of the function:

The domain of the function corresponds to the time during which the depreciation takes place. The company depreciates the computer over 3 years, so the age $ x $ must be between 0 and 3 years (inclusive). Therefore, the domain is:

\[
\text{Domain: } [0, 3]
\]

### (c) Graphing the linear function:

The function $ V(x) = 2400 - 800x $ is a straight line with:
- **Y-intercept** $ V(0) = 2400 $, which represents the initial value.
- **X-intercept** at $ x = 3 $, where the book value drops to $ 0 $.

Let me sketch the graph for you.

### (d) Book value after 2 years:

To find the book value after 2 years, substitute $ x = 2 $ into the linear function:

\[
V(2) = 2400 - 800(2) = 2400 - 1600 = 800
\]

So, the book value after 2 years is **\$800**.

### (e) When will the computer have a book value of \$1600?

We can solve for $ x $ by setting $ V(x) = 1600 $:

\[
1600 = 2400 - 800x
\]

Now solve for $ x $:

\[
1600 - 2400 = -800x
\]
\[
-800 = -800x
\]
\[
x = 1
\]

So, the computer will have a book value of **\$1600** after 1 year.

---

Would you like the graph or further details on any step?

Here are 5 related questions:
1. How does straight-line depreciation compare to other methods like declining balance?
2. What would the function look like if the depreciation were over 5 years instead of 3?
3. How would the function change if the initial value of the computer were \$3000?
4. What happens to the graph if the depreciation rate is not constant?
5. How does the domain affect the graph of the depreciation function?

**Tip:** Always ensure the domain reflects the useful life of the asset in straight-line depreciation problems.

Suppose that a company has just purchased a new computer for $2400. The company chooses to depreciate using the straight-line method for 3 years.
(a) Write a linear function that expresses the book value V of the computer as a function of its age x.
(b) What is the domain of the function found in part (a)?
(c) Graph the linear function.
(d) What is the book value of the computer after 2 years?
(e) When will the computer have a book value of $1600?

Learn how to write a linear function to express the book value of a computer purchased for $2400, using straight-line depreciation over 3 years. This guide includes the domain, graph, and a step-by-step solution to find the book value after 2 years and when it will reach $1600.

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