Problem 92 - Essential Calculus

Question

A company buys a machine for \(\$ 225,000\) that depreciates at a rate of \(30 \%\) per year. Find a formula for the value of the machine after \(n\) years. What is its value after 5 years?

Step-by-Step Solution

Answer

The value of the machine after 5 years is calculated to be approximately \$55,873.22.

Step 1: Setting up the Depreciation Equation

Since annual depreciation is 30%, the value that remains each year is \(70\%\) or \(0.7\) of the previous year. We can express this as a formula where \(P\) is the initial price of the machine and \(n\) is the number of years: \(V = P * (0.7)^n \).

Step 2: Substitute Initial Value

We substitute the given initial value \(P = \$225,000\) into the formula to get the equation for the machine's value after \(n\) years: \(V = 225000 * (0.7)^n \).

Step 3: Find the Machine's Value After 5 Years

Now substitute \(n=5\) into the formula: \(V = 225000 * (0.7)^5 \). Calculate to find the value of the machine after 5 years.

Key Concepts

Exponential Depreciation

When it comes to the financial aspect of assets, understanding how their value decreases over time is crucial. Exponential depreciation is a method used to model the decrease in the value of an asset over time. It's predicated on the premise that the value decreases at a consistent percentage rate each period, leading to a smoother and more realistic depreciation curve compared to methods that assume a fixed amount of depreciation.With this method, as time progresses, the absolute amount of depreciation lessens, since it's a fixed rate applied to a decreasing value. This contrasts with methods like straight-line depreciation, where the amount of depreciation remains constant throughout the asset's life. The concept is very important for students, notably in fields like accounting and finance, where they must understand how to evaluate an asset's current and future worth.

Depreciation Rate

The depreciation rate is essentially the percentage rate at which an asset diminishes in value per time period. It's a pivotal factor in the exponential depreciation model. For the company's machine mentioned in the exercise, the depreciation rate is given as 30% annually, which means the machine loses 30% of its value each year.Understanding the depreciation rate is critical because it helps in planning for asset replacement, tax deductions, and financial reporting. By knowing this rate, one can anticipate when an asset will no longer be financially beneficial to keep and maintain. In educational terms, grasping the concept of depreciation rate aids students in solving problems related to asset management and valuation.

Exponential Decay Formula

The exponential decay formula is integral to calculating the depreciation of an asset over time. It's represented by the equation \(V = P \cdot (1 - r)^n\), where \(V\) is the final value of the asset after \(n\) periods, \(P\) is the initial principal balance (initial value of the asset), \(r\) is the rate of depreciation per period, and \(n\) is the number of time periods.In the context of the example, the machine with an initial value of $225,000 and an annual depreciation rate of 30%, the formula would be applied as \(V = 225000 \cdot (0.7)^n\). After 5 years, you would calculate the value by substituting 5 for \(n\), resulting in a lower value each year due to the effect of exponential depreciation. It's crucial for students to understand not just how to plug numbers into this formula, but the implications it has in real-world situations, such as the long-term financial planning for assets.