Q. 46 - Precalculus Enhanced with Graphing Utilities

Question

Straight-line Depreciation Suppose that a company has just purchased a new machine for its manufacturing facility for \(120,000. The company chooses to depreciate the machine using the straight-line method over 10 years. 

(a) Write a linear model that expresses the book value V of the machine 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 machine after 4 years? 

(e) When will the machine have a book value of \)72,000? 

Step-by-Step Solution

Answer

Part (a) The function is $V\left(x\right)=-12000x+120000$.

Part (b) The domain is $\left\{x|0\le x\le 10\right\}$.

Part (c) The graph is 

Part (d) The book value of the machine after 4 years is $72000. 

Part (e) After 4 year a book value of the computer will be  $ 72000.

Part (a) Step 1. Given information.

A company has just purchased a new machine for its manufacturing facility for $120,000. It chooses to depreciate the machine using the straight-line method over 10 years. 

Part (a) Step 2 Explanation.

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

The original value of Machine as $V\left(0\right)$   is $120000. This y-intercept of the linear function.

The depreciation period is 10 years.

The depreciation per year is $\frac{120000}{10}=12000$ So, slope is -12000.

The linear function that represents the book value V of the machine as a function of its age x  is $V\left(x\right)=-12000x+120000$

Part (b) Step 1 Explanation.

Find the implied domain of the function.

The depreciation period of machine choose by company is 10 years.
 so, x varies from 0 to 10.

The implied domain of the function is $\left\{x|0\le x\le 10\right\}$. 

Part (c) Step 1 Explanation.

The graph of the linear function $v\left(x\right)=-12000x+120000$ on domain $\left[0,3\right]$ is as follows.

Part (d) Step 1 Explanation.

Find the book value of the machine after 4 years.

We have $x=4$, find $V\left(x\right)$.

$$\begin{gathered} V\left(x\right)=-12000x+120000 \\ =12000\left(4\right)+120000 \\ =72000 \end{gathered}$$

The book value of the machine after 4 years is $72000. 

Part (e) Step 1 Explanation.

Find by when a book value of the machine will be of $72,000 .

We have $V\left(x\right)=72000$ Find x

$$\begin{gathered} V\left(x\right)=-12000x+120000 \\ 72000=-12000x+120000 \\ -48000=-12000x \\ 4=x \end{gathered}$$

After 4 year a book value of the computer will be  $ 72000.