Chapter 3.1, Problem 46AYU

(a)

To determine

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

Answer to Problem 46AYU

The linear function that represents the book value is $V\left(x\right)=-12000x+120,000$

Explanation of Solution

Given:

  $T\left(p\right)=0.40\left(p-136.5\right)$

Calculation:

Let $V\left(x\right)$ represent the book value of the machine after x years. Then, $V\left(0\right)$ will represent the original price of the machine. From the given information, $V\left(0\right)=120,000$ . The y -intercept of this function is $120,000.

It is given that the value of the machine gets depreciated over 10 years. So, the value is depreciated by $\frac{\$120,000}{\$10}$ or $12,000 per year. From this, the slope of the function as -12000.

Thus, the linear function that represents the book value is $V\left(x\right)=-12000x+120,000$

Conclusion:

Therefore, the linear function that represents the book value is $V\left(x\right)=-12000x+120,000$

(b)

To determine

To graph: the linear function

Answer to Problem 46AYU

An additional point is $\left(1,108,000\right)$ .

Explanation of Solution

Calculation:

In part (a), the equation $V\left(x\right)=-12000x+120,000$ has - 12000 as the slope, and 120,000 as the y -intercept.

Now, in order to plot the graph, take the y -intercept $\left(0,120,000\right)$ as one point. Since the slope of the line is -12000, move 1 unit horizontally to the right and 12000 units vertically downwards to find an additional point. Thus, an additional point is $\left(1,108,000\right)$ .

Plot the points on a coordinate a plane, and join them using a straight line.

  

Conclusion:

Therefore, an additional point is $\left(1,108,000\right)$ .

(c)

To determine

To find: the book value of the machine after 4 years

Answer to Problem 46AYU

The book value of the machine after 4 years will be $72,000.

Explanation of Solution

Calculation:

Substitute 4 for x in $V\left(x\right)=-12000x+120,000$

V(4) = -12000(4) + 120,000

Evaluate

  $\begin{array}{l}V\left(4\right)=-48000+120,000\\ =72,000\end{array}$

Thus, the book value of the machine after 4 years will be $72,000.

Conclusion:

Therefore, the book value of the machine after 4 years will be $72,000.

(d)

To determine

To discuss: the machine have a book value of $72000

Answer to Problem 46AYU

The machine will have a book value of $72000 after 4 years.

Explanation of Solution

Calculation:

Write the answer directly by using the result from part (c).

The machine will have a book value of $72000 after 4 years.

Conclusion:

Therefore, the machine will have a book value of $72000 after 4 years.