Problem 72 - Algebra and Trigonometry

Question

Depreciation A small business buys a computer for \(\$ 4000\) . After 4 years the value of the computer is expected to be \(\$ 200\) . For accounting purposes the business uses linear depreciation to assess the value of the computer at a given time. This means that if \(V\) is the value of the computer at time \(t\) then a linear equation is used to relate \(V\) and \(t\) (a) Find a linear equation that relates \(V\) and \(t\) (b) Sketch a graph of this linear equation. (c) What do the slope and \(V\) -intercept of the graph represent? (d) Find the depreciated value of the computer 3 years from the date of purchase.

Step-by-Step Solution

Answer

Equation: \( V = -950t + 4000 \). Slope: -950 (depreciation/year), Intercept: 4000 (initial value). Value after 3 years: \$1150.

Step 1: Identify the Points

The computer's value decreases from \( \\(4000 \) to \( \\)200 \) over 4 years. We can represent these values as points: at \( t = 0 \), \( V = 4000 \); at \( t = 4 \), \( V = 200 \).

Step 2: Find the Slope

The formula for the slope (\( m \)) of a line passing through two points \((t_1, V_1)\) and \((t_2, V_2)\) is \( m = \frac{V_2 - V_1}{t_2 - t_1} \). Substituting the given points, \( m = \frac{200 - 4000}{4 - 0} = \frac{-3800}{4} = -950 \).

Step 3: Write the Linear Equation

Using the slope \( m = -950 \) and the point-slope formula \( V - V_1 = m(t - t_1) \), substitute \((t_1, V_1) = (0, 4000)\) to find the linear equation: \( V - 4000 = -950(t - 0) \). Simplifying, we get the equation \( V = -950t + 4000 \).

Step 4: Sketch the Graph

To sketch the graph, plot the points \((0, 4000)\) and \((4, 200)\) and draw a line through them. This line represents the equation \( V = -950t + 4000 \).

Step 5: Interpret the Slope and Intercept

The slope \(-950\) represents the rate of depreciation per year, meaning the computer loses \( \\(950 \) in value each year. The \( V \)-intercept (\( \\)4000 \)) is the initial value of the computer at purchase (\( t = 0 \)).

Step 6: Compute Value After 3 Years

Substitute \( t = 3 \) into the linear equation: \( V = -950(3) + 4000 \). Calculating gives \( V = -2850 + 4000 = 1150 \).

Key Concepts

Linear Equations

Linear equations are mathematical expressions that create a straight line when graphed on a coordinate plane. They are typically written in the format \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept. In the context of depreciation, the value of an item, \( V \), can be expressed as a linear equation with respect to time, \( t \). This is illustrated as \( V = -950t + 4000 \), where the equation depicts how the value changes over time with a declining trend.
Linear depreciation assumes a constant rate of decrease, which simplifies calculations and forecasts for businesses. It is particularly useful for items whose value declines steadily over time, like computers, vehicles, or machinery. Understanding the linear relationship between time and value helps in making informed financial decisions about assets.
The key components of the linear equation are the slope and the intercept, which play crucial roles in interpreting the depreciation trend.

Slope Interpretation

The slope of a linear equation indicates the rate at which one variable changes with respect to another. In our example of computer depreciation, the slope is \(-950\). This means that for every year, the computer's value decreases by \(\$950\).

• The negative sign shows a decrease in value, a common occurrence in depreciation scenarios.
• The magnitude (\(950\)) provides a quantitative measure of how fast the depreciation occurs.

The slope is crucial for businesses aiming to understand the longevity of their assets and plan replacements or upgrades. It allows for predicting future asset values simply and effectively. By having a clear understanding of the depreciation rate, businesses can budget for future purchases or plan financial strategies accordingly.

Graph Sketching

Graphing a linear equation involves plotting points on a coordinate grid and drawing a line through them. For the depreciation example, we graph the line \( V = -950t + 4000 \).
Start by plotting key points:

• The initial value point \((0, 4000)\) indicated at the time of purchase.
• The depreciated value point \((4, 200)\) after four years.

Connect these points with a straight line to illustrate the depreciation over time.
Graph sketching visually represents the relationship between time and value, making it easier to understand how quickly an asset depreciates. It also assists in estimating values within the timeframe, for example, interpolating the value for 1, 2, or even 3 years after the purchase, providing a clear visual indicator of the asset's declining value trajectory.

Value Assessment

Assessing the value of an asset at a specific point in time is key to strategic financial planning. By using the depreciation equation \( V = -950t + 4000 \), we can substitute any time, \( t \), to find the computer's value. For instance, to determine the value after 3 years, substitute \( t = 3 \) into the equation:
\[ V = -950(3) + 4000 = 1150 \]
Thus, three years post-purchase, the computer's value drops to \(\$1150\).
This calculation method helps businesses monitor asset value for insurance, resale, or warranty purposes. It offers an easy way to appraise an asset and make informed decisions about its management. Moreover, understanding this concept is essential for ensuring accurate accounting records and providing a clear financial position.