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Chapter 1: Problem 22

In 2005, National Textile installed a new machine in one of its factories at a cost of \(\$ 250,000\). The machine is depreciated linearly over 10 yr with a scrap value of \(\$ 10,000\). a. Find an expression for the machine's book value in the \(t\) th year of use \((0 \leq t \leq 10)\). b. Sketch the graph of the function of part (a). c. Find the machine's book value in 2009 . d. Find the rate at which the machine is being depreciated.

Short Answer

Expert verified
The machine's book value in the t-th year is given by the linear function \(BV(t) = -24,000t + 250,000\). The graph of this function is a line connecting the points (0, 250,000) and (10, 10,000). In 2009, the machine's book value is $154,000, and the rate of depreciation is $24,000 per year.

Step by step solution

01

Understand the given information

The machine has an initial cost of \(250,000 and is depreciated linearly over 10 years, with a scrap value of \)10,000 after 10 years.
02

Find the linear function for depreciation

Since the machine is depreciated linearly, the book value in the t-th year can be represented by a linear function: \(BV(t) = m \cdot t + b\) where - \(BV(t)\) is the book value in the t-th year - t is the number of years (0 鈮� t 鈮� 10) - m is the slope of the linear function (depreciation rate) - b is the initial book value of the machine (when t = 0) We know that initially, the machine's book value is \(250,000 (\(b = 250,000\)) and after 10 years, the book value is \)10,000: \(BV(10) = m \cdot 10 + 250,000 = 10,000\) Now, we can solve for the slope of the linear function (m) and find the equation of the function.
03

Solve for the depreciation rate (slope m)

Using the equation from Step 2: \(10m + 250,000 = 10,000\) Solving for m: \(10m = -240,000\) \(m = -24,000\) Now the equation of the function is: \(BV(t) = -24,000t + 250,000\)
04

Sketch the graph of the function

To sketch the graph, plot the points (0, 250,000) and (10, 10,000) on a coordinate plane. Draw a line connecting these two points. The line represents the machine's book value as a function of time.
05

Find the book value in 2009

In 2009, the machine has been in use for (2009 - 2005) = 4 years. Plug t = 4 in the equation: \(BV(4) = -24,000(4) + 250,000\) \(BV(4) = -96,000 + 250,000\) \(BV(4) = \$154,000\) The machine's book value in 2009 is $154,000.
06

Find the rate of depreciation

The rate of depreciation is the slope (m) of the linear function, which we found in Step 3: The rate of depreciation is \(\$24,000\) per year.

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