Problem 64 - Brief Applied Calculus

Question

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 newspaper buys a printing press for $$\$ 800,000$$ and estimates its useful life to be 20 years, after which its scrap value will be $$\$ 60,000$$. Use the formula above Exercise 63 to find a formula for the value \(V\) of the press after \(t\) years, for \(0 \leq t \leq 20\) b. Use your formula to find the value of the press after 10 years. c. Graph the function found in part (a) on a graphing calculator on the window \([0,20]\) by \([0,800,000] .\) [Hint: Use \(x\) instead of \(t\).]

Step-by-Step Solution

Answer

a. \( V(t) = 800,000 - 37,000t \); b. \( V(10) = 430,000 \); c. Graph is a line from (0, 800,000) to (20, 60,000).

Step 1: Understand the Formula

For straight-line depreciation, the formula to find the value of an asset after \( t \) years is given by:\[ \text{Value} = (\text{Price}) - \left( \frac{(\text{Price}) - (\text{Scrap value})}{(\text{Useful lifetime})} \right) \cdot t \] where \( t \) is the number of years, Price is the original cost of the asset, Scrap value is the value at the end of its useful life, and Useful lifetime is the total years expected to use the asset.

Step 2: Plug Numbers into the Formula

Given:- Price \( = \\(800,000 \)- Scrap value \( = \\)60,000 \)- Useful lifetime \( = 20 \text{ years} \)Substitute these values into the formula:\[ V(t) = 800,000 - \left( \frac{800,000 - 60,000}{20} \right) \cdot t \]

Step 3: Simplify the Formula

First, calculate the depreciation rate:\[ \text{Depreciation rate} = \frac{800,000 - 60,000}{20} = \frac{740,000}{20} = 37,000 \]This means the printing press depreciates by \( \$37,000 \) per year.Plug this back into the equation:\[ V(t) = 800,000 - 37,000t \]This is the formula for the value \( V \) of the press after \( t \) years.

Step 4: Find the Value after 10 Years

Use the formula found in Step 3, \( V(t) = 800,000 - 37,000t \), and substitute \( t = 10 \):\[ V(10) = 800,000 - 37,000 \times 10 = 800,000 - 370,000 = 430,000 \]Therefore, the value of the press after 10 years is \( \$430,000 \).

Step 5: Graph the Function

To graph the function on a calculator, use the equation \( V(x) = 800,000 - 37,000x \). Set your graphing calculator window from \([0, 20]\) for \( x \) (years), and \([0, 800,000]\) for \( V(x) \) (value in dollars). The graph will be a straight line starting at \( (0, 800,000) \) and ending at \( (20, 60,000) \).

Key Concepts

Asset Depreciation

When we talk about asset depreciation, we're referring to the gradual reduction in the value of a tangible asset over time. This happens due to factors like wear and tear, technological obsolescence, or market shifts. Imagine buying a new car; its value starts to drop the moment you drive it off the lot!

For businesses, understanding asset depreciation is crucial. This is not just about acknowledging the lost value but also about accurately reflecting this change in value on financial statements. The concept helps businesses budget for replacements, claim tax deductions, and make informed decisions about asset management.

• Types of Depreciation Methods: Straight-line depreciation, declining balance, and sum-of-the-years'-digits, to name a few.
• Importance: It affects the balance sheet, reduces tax liability, and influences investment decisions.

Distinguishing between the different methods of depreciation can impact how a company's performance and asset value are perceived by investors and stakeholders.

Mathematical Modeling

Mathematical modeling is a powerful tool used to represent real-world phenomena through mathematical expressions. It allows us to predict future outcomes and understand complex systems. In this exercise, the mathematical model is expressed as a formula for straight-line depreciation.

The formula used is:
\[V(t) = ( ext{Price}) - rac{( ext{Price}) - ( ext{Scrap value})}{( ext{Useful lifetime})} imes t\]Here,

• Parameters: Each element like Price, Scrap value, and Useful lifetime are inputs into the model, which affect the outcome.
• Outcome: The output is the value of the asset after a specific number of years.

Mathematical models simplify complex processes into digestible formulas, which can then be used for prediction and optimization in business settings. These models are highly valuable in finance for modeling asset behavior and depreciation trends.

Business Mathematics

Business mathematics is the study of mathematical formulas and techniques used in commercial enterprises. It's like the backbone of everyday business operations, helping executives and managers solve real-life problems efficiently. Straight-line depreciation, as shown here, is just one of the many applications of business mathematics.

By using formulas and equations, businesses can analyze performance, predict future financial states, and make informed decisions. The process involves:

• Problem Definition: Clearly defining what needs to be solved or understood, like the depreciation of an asset.
• Using Formulas: Applying appropriate mathematical formulas, such as the depreciation formula, to find solutions.
• Analysis: Interpreting the results to provide insights for decision-making.

Understanding business mathematics empowers companies to adopt more strategic financial practices, like budgeting for asset replacement or evaluating return on investment. By applying these principles, businesses ensure they remain competitive and financially sound.