Problem 79 - A Graphical Approach to College Algebra

Question

Depreciation in Value Each year a machine loses \(20 \%\) of the value it had at the beginning of the year. Find the value of the machine at the end of 6 years if it cost \(\$ 100,000\) new.

Step-by-Step Solution

Answer

The machine's value will be approximately $26,214.40 after 6 years.

Step 1: Understand the Depreciation Model

The machine depreciates by 20% each year. This means each year it retains 80% of its previous year's value, as 100% - 20% = 80%. We can express this as a decay factor of 0.8. Therefore, each year's value is 80% of the previous year's value.

Step 2: Formulate the Depreciation Equation

Let the initial value of the machine be \( V_0 = 100,000 \). The value at the end of each year can be modeled using the equation:\[V_n = V_0 \times (0.8)^n\]where \( V_n \) is the value of the machine after \( n \) years.

Step 3: Calculate the Value After 6 Years

To find the value after 6 years, substitute \( n = 6 \) into the equation from Step 2:\[V_6 = 100,000 \times (0.8)^6\]

Step 4: Compute the Final Value

Calculate \((0.8)^6\):\[(0.8)^6 \approx 0.262144\]Multiply this by the initial value:\[100,000 \times 0.262144 = 26,214.40\]

Step 5: Interpret the Result

After evaluating the expression, the machine will be valued at approximately \( \$26,214.40 \) at the end of 6 years.

Key Concepts

Exponential Decay

When we talk about exponential decay, we are referring to the process of decrease at a consistent rate, especially in cases like depreciation. In this exercise, the machine's value decreases by a fixed percentage each year, specifically by 20%. The concept of exponential decay is vital in understanding how things lose value or amount over time. Here, rather than losing a constant number each year, it loses a constant percentage. Whenever something decreases at a constant rate per unit time, it's best described with exponential decay. In this context, the machine retains 80% of its previous year's value (since it loses 20%), leading to a decay factor. This decay factor is 0.8, because 100% minus 20% leaves 80%, or 0.8 in decimal form. Thus, the machine's value is consistently multiplied by 0.8 every year, demonstrating exponential decay.

Value Calculation

To calculate the value of anything that undergoes exponential decay, we use specific methods and formulas. For the machine in question, we start by identifying the initial value, which is given as $100,000. This is our starting point from which all subsequent values will be calculated.Next, we determine a decay factor, which in this problem is 0.8, and it represents the percentage of the value retained each year. The equation used for calculating the machine's value after a specified number of years is \( V_n = V_0 \times (0.8)^n \). Here, \( V_n \) is the unknown value after \( n \) years, while \( V_0 \) is the initial value.Using this formula, we substitute \( n = 6 \) to find the machine's value after 6 years. By calculating the expression \( (0.8)^6 \), we find it equals approximately 0.262144. Multiplying this result by the initial value gives us the depreciated value at that time.

Algebraic Modeling

Algebraic modeling is a technique that utilizes algebraic expressions and equations to describe real-world problems. When working with depreciation, algebraic modeling becomes a handy tool. It helps in forming an equation that can predict future values. In this case, the machine's depreciation is modeled using the formula \[ V_n = V_0 \times (0.8)^n \]. This equation allows us to make predictions about the machine's value at any given point, assuming the depreciation rate remains constant.The beauty of algebraic modeling lies in its ability to abstract complex real-world phenomena into simple, manageable equations. Here, the number of years, decay factor, and initial value combine to make it easy to calculate and understand how the value changes over time. Using this equation, students can plug in different values to see how varying conditions affect the end result.