Chapter 3.1, Problem 45AYU

(a)

To determine

To write: thelinear model that expresses the book value V of the computer as a function of its age x .

Answer to Problem 45AYU

The function is given by the equation $V\left(x\right)=-1000x+3000$ .

Explanation of Solution

Given:

Calculation:

Let $V\left(x\right)$ represent the book value of a computer after x years. Then, $V\left(0\right)$ represents the original price of the computer. So, $V\left(0\right)=3000$ . The y -intercept of the function is 3000.

Now, the value of the computer is depreciated over 3 years. So, the value is depreciated by $\frac{3000}{3}$ or 1000 per year.

Since each computer is depreciated by $1000 per year, the slope of the function is $-1000$ .

Therefore, the function is given by the equation $V\left(x\right)=-1000x+3000$ .

Conclusion:

Therefore, the function is given by the equation $V\left(x\right)=-1000x+3000$ .

(c)

To determine

To graph: the linear function.

Answer to Problem 45AYU

Thegraph is drawn.

Explanation of Solution

Calculation:

Compare the equation $V\left(x\right)=-1000x+3000$ with the standard linear equation $y=mx+b$ to find the slope and the y -intercept. Thus, $-1000$ is the slope and 3000 is the y -intercept.

To plot the graph, take the y -intercept $\left(0,3000\right)$ as one point. Since the slope of the line is $-1000$ , move 1 unit horizontally to the right and 1000 units vertically downwards to find an additional point. Thus, an additional point is $\left(1,2000\right)$ .

The graph can be plotted as follows.

  

Conclusion:

Therefore, thegraph is drawn.

(d)

To determine

To find: the book value of the computer after 2 years.

Answer to Problem 45AYU

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

Explanation of Solution

Calculation:

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

  $\begin{array}{l}V\left(2\right)=-1000\cdot 2+3000\hfill \\ \begin{array}{l}=-2000+3000\\ =1000\end{array}\hfill \end{array}$

Conclusion:

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

(e)

To determine

To find: when will the computer have a book value of $2000.

Answer to Problem 45AYU

The book value of the computer will be $2000 after 1year.

Explanation of Solution

Calculation:

Substitute 2000 for $V\left(x\right)$ in $V\left(x\right)=-1000x+3000$ .

  $2000=-1000x+3000$

Subtract 3000 from both the sides.

  $\begin{array}{l}2000-3000=-1000x+3000-3000\hfill \\ -1000=-1000x\hfill \end{array}$

Swap the sides of the equation.

  $-1000x=-1000$

Divide both the sides by − 1000.

  $\begin{array}{l}\frac{-1000x}{-1000}=\frac{-1000}{-1000}\\ x=1\end{array}$

Conclusion:

Therefore, the book value of the computer will be $2000 after 1year.