Problem 63 - Brief Applied Calculus

Question

63-64. BUSINESS: Straight-Line Depreciation Straight-line depreciation is a method for estimating the value of an asset (such as a piece of machinery) as it loses value ( \({ }^{\prime \prime}\) depreciates" \()\) through use. Given the original price of an asset, its useful lifetime, and its scrap value (its value at the end of its useful lifetime), the value of the asset after \(t\) years is given by the formula: $$ \begin{aligned} \text { Value }=(\text { Price })-&\left(\frac{(\text { Price })-(\text { Scrap value })}{(\text { Useful lifetime })}\right) \cdot t \\ & \text { for } 0 \leq t \leq(\text { Useful lifetime }) \end{aligned} $$ a. A farmer buys a harvester for $$\$ 50,000$$ and estimates its useful life to be 20 years, after which its scrap value will be $$\$ 6000$$. Use the formula above to find a formula for the value \(V\) of the harvester after \(t\) years, for \(0 \leq t \leq 20\). b. Use your formula to find the value of the harvester after 5 years. c. Graph the function found in part (a) on a graphing calculator on the window \([0,20]\) by \([0,50,000]\). [Hint: Use \(x\) instead of \(t\).]

Step-by-Step Solution

Answer

a. The formula is \(V = 50000 - 2200t\). b. After 5 years, the value is $39,000. c. Graph the equation on calculated window to see depreciation.

Step 1: Understanding the Problem

We need to determine the formula for the straight-line depreciation of a harvester with an original price of $50,000, a scrap value of $6,000, and a useful life of 20 years. The formula for depreciation is provided.

Step 2: Setting Up the Formula

Using the given depreciation formula \[ V = \text{Price} - \left(\frac{\text{Price} - \text{Scrap value}}{\text{Useful lifetime}}\right) \cdot t \], we substitute the known values: Price = \(50,000, Scrap Value = \)6,000, and Useful Lifetime = 20.

Step 3: Simplifying the Formula

Substitute the known quantities into the formula:\[ V = 50000 - \left( \frac{50000 - 6000}{20} \right) \cdot t \].Calculate the depreciation rate: \[ \frac{50000 - 6000}{20} = \frac{44000}{20} = 2200 \].This means the harvester loses $2200 in value each year.

Step 4: Finalizing the Formula

Insert the annual depreciation rate into the formula:\[ V = 50000 - 2200t \].This is the formula for the value of the harvester over time, where \(V\) is the value after \(t\) years.

Step 5: Calculating Value After 5 Years

Use the formula \(V = 50000 - 2200t\) and substitute \(t = 5\):\[ V = 50000 - 2200 \cdot 5 \].Calculate:\[ V = 50000 - 11000 = 39000 \].After 5 years, the harvester is valued at $39,000.

Step 6: Graphing the Depreciation Function

To graph the depreciation over time, use the function \(V = 50000 - 2200x\) on a graphing calculator where \(x\) represents years. Set the window from \([0, 20]\) for \(x\) and from \([0, 50000]\) for \(V\). Observe the linear decrease in value over time.

Key Concepts

Asset Valuation

When a business acquires a physical asset, such as machinery or equipment, understanding its value over time is crucial. The term **asset valuation** refers to the process of determining the worth of an asset. This is important because assets lose value as they age and are used, a process known as depreciation. Understanding depreciation helps businesses make informed financial decisions.
To effectively manage assets, businesses need to determine both the current value and estimated future value of these possessions. This involves considering factors such as the initial purchase price, any installation costs, and potential revenue generation. Accurate asset valuation leads to better budgeting, tax reporting, and asset management.
In the case of the harvester example, knowing that it initially costs $50,000 and devaluates over 20 years to a scrap value of $6,000 helps paint a clear picture of its diminishing worth. Evaluating assets this way allows for strategic planning around maintenance, replacement, and operational investments.

Depreciation Formula

The **depreciation formula** is a mathematical method to express how much an asset's value decreases over time. The specific approach called *straight-line depreciation* is one of the simplest and most commonly used.
Here's why it's effective:

• It provides a clear, linear method of calculating asset value loss, making it easy to forecast future worth.
• This method assumes an equal amount of depreciation each year, which simplifies accounting and financial analysis.

The formula itself is expressed as:\[ V = ext{Price} - rac{ ext{Price} - ext{Scrap value}}{ ext{Useful lifetime}} imes t \]Where:

• \(V\) is the asset's value after \(t\) years.
• \( \text{Price} \) is the initial purchase price of the asset.
• \( \text{Scrap value} \) is the expected value at the end of its useful life.
• \( \text{Useful lifetime} \) is the total time the asset is expected to be useful.

In our example, this becomes \( V = 50000 - 2200t \), indicating a $2200 annual value decrease. This straightforward calculation aids businesses in managing their long-term financial plans.

Useful Lifetime Calculation

Determining the **useful lifetime** of an asset is vital for calculating depreciation. It defines the period over which an asset is expected to be functional and contribute to business operations. Assigning a realistic useful life impacts financial reporting and tax obligations.
To determine the useful lifetime:

• Consider the manufacturer's recommendations and industry standards.
• Factor in the expected usage and the environment in which the asset operates.
• Review any technological advancements that might render the asset obsolete sooner.

For example, the harvester in our problem has a useful lifetime of 20 years. To assess this, one would take into account typical wear and tear, expected annual workload, and maintenance practices.
Proper calculation of an asset's useful life ensures accurate depreciation measurement, allowing for the cost-effective use of business resources. It helps businesses schedule maintenance, plan for replacements, and allocate funds efficiently for capital investments.