Question

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 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 \(\$ 72,000 ?\)

Short answer

The linear model is \[V(x) = 120,000 - 12,000x\], the domain is \[0 \leq x \leq 10\]. After 4 years, the value is \)72,000 and it will be $72,000 when the machine is 4 years old.

Step-by-step solution

Understanding the Problem

The company bought a machine for \(120,000 and will use the straight-line depreciation method over 10 years. Need to find a model for book value, the domain, create a graph, find the value after 4 years, and determine when the book value is \)72,000.

Find the Depreciation Rate

The depreciation rate is the amount by which the machine's value decreases each year. Using the straight-line method: \[\text{Depreciation Rate} = \frac{\text{Initial Value} - \text{Final Value}}{\text{Number of Years}} = \frac{120,000 - 0}{10} = 12,000\]

Write the Linear Model

The book value function can be expressed as: \[V(x) = 120,000 - 12,000x\] where `x` is the age of the machine in years.

Determine the Domain

Since the machine depreciates over 10 years, the domain (age of the machine) is: \[0 \leq x \leq 10\]

Graph the Function

Plot the linear function \[V(x) = 120,000 - 12,000x\] on a coordinate plane with x (age in years) on the horizontal axis and V(x) (book value) on the vertical axis. The graph is a straight line starting from (0, 120,000) to (10, 0).

Calculate Book Value After 4 Years

Substitute `x = 4` into the function: \[V(4) = 120,000 - 12,000 \cdot 4 = 120,000 - 48,000 = 72,000\] The book value after 4 years is \$72,000.

Determine When Book Value is \(72,000

Set V(x) to 72,000 and solve for `x`: \[72,000 = 120,000 - 12,000x\] \[12,000x = 120,000 - 72,000\] \[x = \frac{48,000}{12,000} = 4\] The machine will have a book value of \$72,000 after 4 years.

Key concepts

straight-line depreciation

Straight-line depreciation is one of the simplest and most commonly used methods to allocate the cost of a tangible asset over its useful life. It spreads the cost evenly across each year of the asset's expected utility. For instance, suppose a company purchases a machine for \$120,000 with an estimated useful life of 10 years.
The value at the end of its useful life is assumed to be \$0, which makes it easier to calculate the annual depreciation amount. It is computed using the formula:
\[ \text{Annual Depreciation} = \frac{\text{Initial Value} - \text{Final Value}}{\text{Number of Years}} \]
Plugging in our given values: \( \frac{120,000 - 0}{10} = 12,000 \). Hence, the machine loses \$12,000 in value each year.

This method is ideal for assets that lose their value steadily over time, such as machinery, buildings, and vehicles.

Knowing how to calculate straight-line depreciation is crucial for accurate financial reporting and tax purposes.

linear equations

Linear equations are foundational in mathematics, representing relationships with a constant rate of change. These equations are characterized by variables that appear to the first power and produce a straight line when graphed.

For the given machine depreciation problem, the linear model describes how the book value of the machine decreases over time. The equation is given as:
\[ V(x) = 120,000 - 12,000x \]
In this equation:

• \text{\textbf{V(x)}}: book value after \text{x} years
• \text{\textbf{x}}: age of the machine in years

The slope \textbf{12,000} represents the depreciation rate, illustrating how much value the machine loses each year. The y-intercept (\(120,000\)) represents the initial cost of the machine.

Linear equations such as this are essential in various fields, including economics, physics, and engineering. They help to model and predict behaviors over time. Graphing these equations provides a visual understanding, with the slope indicating the rate of change, and the line confirming the linear relationship.

book value

Book value refers to the current value of an asset as recorded on the company's balance sheet. It is calculated as the initial cost of the asset minus the accumulated depreciation.

For our machine, its book value at any year can be found using the linear depreciation model:
\[ V(x) = 120,000 - 12,000x \]
Here are a few key points regarding book value:

• \text{\textbf{Initial Cost}}: The purchase price of the machine (\(120,000\)).
• \text{\textbf{Accumulated Depreciation}}: The total depreciation applied to the asset from the time it was purchased until the specified year.
• \text{\textbf{Current Book Value}}: The value of the machine at any given time, obtained by subtracting the accumulated depreciation from the initial cost.

For example, after 4 years, the book value is calculated as:
\[ V(4) = 120,000 - 12,000 \times 4 = 72,000 \]
Thus, the machine is valued at \$72,000 after 4 years.

Understanding book value is essential for assessing an asset's worth over time, aiding in investment decisions, and ensuring proper accounting practices. It is a critical concept in finance and accounting.