Problem 21 - Technical Mathematics with Calculus

Question

A milling machine having a purchase price \(P\) of \(\$ 15,600\) has an annual depreciation \(A\) of \(\$ 1600 .\) Graph the book value \(y\) at the end of each year, for \(t=0\) to 10 years, using the equation \(y=P-A t\)

Step-by-Step Solution

Answer

The initial book value is \$15,600 at \(t=0\). The book value decreases by \$1,600 each year, resulting in a straight line graph with points from (0, 15600) to (10, 400).

Step 1: Understand the Depreciation Equation

The given equation for book value is a linear equation of the form \(y = P - At\), where \(y\) is the book value after \(t\) years, \(P=\$15,600\) is the purchase price, and \(A=\$1,600\) is the annual depreciation. The graph of this equation will be a straight line.

Step 2: Calculate Initial Book Value

Calculate the initial book value at \(t=0\). Substitute \(t=0\) into the book value equation to find the book value at the time of purchase: \(y = 15,600 - 1,600(0) = 15,600\). Plot the point \((0, 15600)\) on the graph.

Step 3: Calculate Book Value Each Year

Calculate the book value for each subsequent year, from \(t=1\) to \(t=10\). Use the given depreciation equation \(y = P - At\) for each year and find the resulting book values. These will be the points \((t, y)\) on the graph.

Step 4: Plot the Points and Draw the Graph

Plot each calculated point on a graph with time \(t\) on the horizontal axis and book value \(y\) on the vertical axis. Once all points are plotted for \(t=0\) to \(t=10\), draw a line through the points to complete the graph.

Key Concepts

Depreciation Calculation

Depreciation is an accounting method used to allocate the cost of a tangible asset over its useful life. In linear depreciation, also known as straight-line depreciation, an asset's cost is reduced equally each year until it reaches its salvage value or is fully depreciated.

For example, consider an asset, like a milling machine, purchased at \( P = \$15,600 \) with an annual depreciation of \( A = \$1,600 \). Using the linear depreciation method, we would subtract the annual depreciation from the asset's purchase price each year to calculate the book value after \( t \) years, using the formula \( y = P - At \). After ten years, we would expect the asset to be fully depreciated, or have a book value that reflects its salvage value if any remains.

Book Value of Assets

The book value of an asset represents its worth as recorded in the accounting records. This value decreases over time as depreciation is applied. Initially, the book value is equal to the purchase price of the asset. Assuming no salvage value, it eventually reaches zero after the asset has been fully depreciated.

Calculating the book value at any given time can be crucial for financial reporting and tax purposes. In our milling machine example with a purchase price of \( \$15,600 \) and annual depreciation of \( \$1,600 \), the book value at the end of year one would be \( \$14,000 \), and it would continue to diminish by \( \$1,600 \) each year thereafter.

Linear Equations in Economics

Linear equations are a staple in economic analysis and represent relationships between variables with a constant rate of change. They are essential tools for modeling economic phenomena such as cost, revenue, supply, and demand.

The equation for depreciation, \( y = P - At \), is a simple linear equation where the slope, \( -A \), indicates the rate at which the book value decreases. In this case, the variable \( y \) is the book value of the asset, \( P \) is the initial purchase price, and \( t \) is the time in years. The application of this linear equation simplifies forecasting future values and clarifying the financial implications of asset depreciation.

Financial Mathematics

Financial mathematics is the field of applied mathematics concerned with financial markets and economic data. It heavily relies on numerical techniques and models to solve various problems, including those involving depreciation calculations.

By understanding concepts such as present value, interest rates, annuities, and of course, depreciation, stakeholders can make more informed financial decisions. The linear nature of the straight-line depreciation method allows for clear-cut and predictable budgeting for asset replacement and can be helpful when planning long-term financial strategies.