Chapter 3.1, Problem 46AYU

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 $\$72,000$?

To determine

To calculate:

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 $\$72,000$?

Answer to Problem 46AYU

Solution:

a. $V(x)=-12000x+120000$

b. The domain of the given function is $[0,10]$.

c. The graph is as shown below:

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

e. The computer will have a book value of $\$72000$ after 4 year.

Explanation of Solution

Given:

The company purchases a new machine for $\$120,000$.

The depreciation is done for 10 years.

Calculation:

a.

Since the value of the computer is depreciated, the average rate of change of the computer is $-\$12000$.

Therefore, the linear function representing the book value of the computer is $V(x)=-12000x+120000$, where $V(x)$ is the value of the machine after each year and 120000 is $V(0)$, which represents the original value of the machine.

b.

The value of $x$ in the given function can be any number between 0 and 10.

Therefore, the domain of the given function is $[0,10]$.

c.

The graph of the linear function is

d.

$V(4)=-12000(4)+120000$

$=72000$

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

e.

$72000=-12000(x)+120000$

$\Rightarrow x=\frac{72000-120000}{-12000}=4$

The computer will have a book value of $\$72000$ after 4 year.