Problem 65 - Applied Calculus

Question

Straight-line depreciation is a method for estimating the value of an asset (such as a piece of machinery) as it loses value ("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. \(V = 50,000 - 2,200t\); b. \$39,000; c. Graph a line from \( (0, 50,000) \) to \( (20, 6,000) \).

Step 1: Understand the problem

We have been given the original price of the harvester as \\(50,000, a useful lifetime of 20 years, and a scrap value of \\)6,000. We need to use the straight-line depreciation formula to create a formula for the harvester's value over time and find out its value after 5 years.

Step 2: Substitute known values into the formula

Using the formula \[\text{Value} = (\text{Price}) - \left(\frac{(\text{Price}) - (\text{Scrap value})}{(\text{Useful lifetime})}\right) \cdot t\], we substitute:\[\text{Price} = 50,000,\ \text{Scrap value} = 6000,\ \text{Useful lifetime} = 20.\] This gives us:\[\text{Value} = 50,000 - \left(\frac{50,000 - 6,000}{20}\right) \cdot t.\]

Step 3: Simplify the formula

Calculate \(50,000 - 6,000 = 44,000\). Then divide by the useful lifetime: \(\frac{44,000}{20} = 2,200\). So the simplified formula becomes:\[\text{Value} = 50,000 - 2,200t.\] This is the depreciation formula for the harvester over time.

Step 4: Calculate the value after 5 years

Substitute \(t = 5\) into the formula:\[\text{Value} = 50,000 - 2,200 \times 5 = 50,000 - 11,000 = 39,000.\]The value of the harvester after 5 years is \$39,000.

Step 5: Describe graphing the function

To graph the function \(V = 50,000 - 2,200x\) using a graphing calculator, set the window to \([0, 20]\) for the x-axis and \([0, 50,000]\) for the y-axis. The graph will be a straight line descending from \\(50,000 when \(x = 0\) to \\)6,000 when \(x = 20\).

Key Concepts

Asset Valuation

When talking about assets, we're discussing their worth or value, which can change over time. In the case of tangible assets like machinery, they tend to lose value as they are used. This process is called depreciation. It's important to understand what constitutes an asset's initial valuation, which includes its original purchase price. You also have to consider other values, like the scrap value, which is what an item is worth at the end of its useful life. To determine an asset's value at any point during its lifespan, we use various methods. In this exercise, we use straight-line depreciation because it offers a simple calculation of how much value the asset loses each year.

Depreciation Formula

The depreciation formula is a valuable tool in financial mathematics. It helps us estimate the decrease in value of an asset each year. For straight-line depreciation, the formula is written as:\[ \text{Value} = (\text{Price}) - \left(\frac{(\text{Price}) - (\text{Scrap value})}{(\text{Useful lifetime})}\right) \times t \]This formula has a few key components:

• Price: The initial cost to acquire the asset.
• Scrap Value: The expected value after the asset's useful life.
• Useful Lifetime: The estimated period the asset will be in service.
• t: The time in years since the asset was purchased.

Plugging these values into the formula allows us to see how much the asset is worth over time, reflecting its gradual decrease in value.

Graphing Depreciation

One way to better understand how an asset's value decreases over time is through graphing. This visual representation helps us see the pattern or trend effortlessly. For straight-line depreciation, the graph is always a straight line. It starts at the initial price of the asset when it's new and ends at its scrap value after its useful life. In our example, the graph begins at $50,000 and moves downwards to $6,000 over 20 years. Using graphing tools, set your x-axis from 0 to the useful lifetime (20 years in this case), and your y-axis from 0 up to the original price. This shows a clear, expected decline over time.

Financial Mathematics

Understanding financial mathematics is key in making informed decisions about asset management. This field combines mathematical methods with financial applications, making processes like depreciation more approachable. Straight-line depreciation is a common application. It simplifies otherwise complicated calculations, allowing you to predict the financial trajectory of assets in a business. In essence, implementing formulas like our depreciation example ensures that businesses can manage their resources wisely. This helps in budgeting and financial planning by providing a predictable outline of future expenses.