Q. 45 - Precalculus Enhanced with Graphing Utilities

Question

Straight-line Depreciation Suppose that a company has just purchased a new computer for \(3000. The company chooses to depreciate the computer using the straight-line method over 3 years.

(a) Write a linear model that expresses the book value V of the computer as a function of its age x. 

(b) What is the implied 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 \)2000? 

Step-by-Step Solution

Answer

Part (a) The function is $V\left(x\right)=-1000x+3000$

Part (b) The implied domain of the function is $\left\{x|0\le x\le 3\right\}$.

Part (c) The graph is 

Part (d) The book value of the computer after 2 years $1000.

Part (e) After 1 year a book value of the computer will be  $2000.

Part (a) Step 1. Given Information.

A company has just purchased a new computer for $3000. The company choose depreciation of the computer using the straight-line method over 3 years. 

Part (a). Step 2. Explanation.

Find a linear model that expresses the book value V of the computer as a function of its age x. 

The original value of computer as $V\left(0\right)$ is $3000. This y-intercept of the linear function.

The depreciation per year is $\frac{3000}{3}=1000$. So, slope is -1000.

The linear function that represents the book value V of the computer as a function of its age x  is $V\left(x\right)=-1000x+3000$.

Part (b). Step 1. Explanation.

Find the implied domain of the function.

The depreciation period of the computer is 3 yrs.

so x varies from0 to 3.

The implied domain of the function is $\left\{x|0\le x\le 3\right\}$.

Part (c). Step 1. Explanation.

the graph of the linear function on domain $\left[0,3\right]$ is as follows.

Part (d). Step 1. Explanation.

Find the book value of the computer after 2 years.

Here, $x=2$

Substitute $x=2$ in $V\left(x\right)=-1000x+3000$.

$$\begin{gathered} V\left(x\right)=-1000x+3000 \\ =-1000\left(2\right)+3000 \\ =1000 \end{gathered}$$ 

The book value of the computer after 2 years is $1000.

Part (e). Step 1. Explanation.

Find by when a book value of the computer will be  $2000.

we have $V\left(x\right)=2000$ Find x.

$$\begin{gathered} V\left(x\right)=-1000x+3000 \\ 2000=-1000x+3000 \\ -1000=-1000x \\ 1=x \end{gathered}$$

After 1 year a book value of the computer will be  $2000 .